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6_Probability and Statistic Russian Papers
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simpanSimpan 6_Probability and Statistic Russian Papers Untuk Nanti
3871704 Culture Change
Journal of Information Science
Philippines in 2030 the Future Demographic
Population Projections 2010-2020
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Methodology for Life Table
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Rowland Et Al JMTE 8(3)
Four Myths of Marriage
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Biblical Encouragement for the Modern Day Wife
Understanding the opposite sex (the male and female single)
A_LETTER_TO_A_DAUGHTER_FINAL_OUTPUT[1].docx
TMC 2012 Program
Russian papers
Selected and Translated by Oscar Sheynin
(C) Oscar Sheynin www.sheynin.de
0. Foreword ...
Part 1. The Early Period
1. Paevsky, V.V. Eulers work in demography. In collected papers Leonard Euler. Moscow
Leningrad, 1935, pp. 103 110
2. Bernoulli, D. On the mean duration of marriages, etc. (1768). Translated from Russian
(History of Statistics), vol. 1. Moscow,
translation in Ptukha, M.V.
1955, pp. 453 464...
3. Ondar, Kh.O. On the works of Davidov in probability theory and on his methodological
views. Istoria i Metodologia Estesvennykh Nauk, vol. 11, 1971, pp. 98 109
4. Gnedenko, B.V. On Ostrogradskys work in the theory of probability. IstorikoMatematich. Issledovania, vol. 4, 1951, pp. 99 123
Part 2. The Petersburg School of the Theory of Probability
5. Bernstein, S.N. Chebyshevs work in the theory of probability (1945). In
(Coll. Works), vol. 4. N.p., 1964, pp. 409 433
6. Usov, N.A. A remark concerning the Chebyshev theorem. Matematich. Sbornik, vol. 2,
1867, pp. 93 95 of second paging
7. Bernstein, S.N. The Petersburg school of the theory of probability. Uchenye Zapiski
Leningradsk. Gosudarstven. Univ., No. 55 (ser. math. sci., No. 10), 1940, pp. 3 11
8. Sheynin, O. On the relations between Chebyshev and Markov. Istoriko-Matematich.
Issledovania, to appear
9. Liapunov, A.M. On the Gauss formula for estimating the measure of precision of
observations. Manuscript, first published by the present translator in Istoriko-Matematich.
Issledovania, vol. 20, 1975, pp. 319 328.
10. Markov, A.A. The law of large numbers and the method of least squares (1899).
(Sel. Works). N.p., 1951, pp. 233 251.
11. Markov, A.A. The extension of the law of large numbers, etc. [1906 (1907)]. Ibidem, pp.
12. Markov, A.A. The extension of the limit theorems, etc. (1908). Ibidem, pp. 365 397.
13. Markov, A.A. An investigation of the general case of trials, etc. (1910). Ibidem, pp. 467
14. Markov, A.A. On the coefficient of dispersion (1916). Ibidem, pp. 525 535
15. Markov, A.A. On the coefficient of dispersion for small numbers. Strakhovoe Obozrenie,
No. 2, 1916, pp. 55 59
16. The correspondence between A.A. Markov and B.M. Koialovich. Archival letters
17. Linnik, Yu.V., et al, A sketch of the work of Markov, etc. In Markov, A.A.
18. Markov, A.A.[, Jr], The biography of A.A. Markov[, Sr]. Ibidem, pp. 599 613.
19. Sheynin, O. On the probability-theoretic heritage of Cournot. Istoriko- Matematich.
Issledovania, vol. 7 (42), 2002, pp. 301 316.
20. Gnedenko, B.V. A review of the current state of the theory of limiting laws of
independent terms. Uchenye Zapiski Tomsk. Gosudarstven. Pedagogich. Inst., No. 1, 1939,
pp. 5 28.
0. Foreword
I am presenting a collection of translations of Russian papers on probability and statistics
important at least for the history of these disciplines. One of Gnedenkos papers included
here treats comparatively new developments, which, however, might be considered as a
continuation of the works of the 19th century. I also included two my own papers one of
which (on Cournot) might seem to be out of the general context, but it describes the work of
an outstanding Western scientist who at the very least influenced both contemporaneous
(Davidov) and later (Chuprov) Russian scholars. Along with the abovementioned work of
Gnedenko, it forms the Supplement to this collection whereas the first few items bearing on
the early period in Russian probability constitute Part 1. The main body of this book is given
over to Part 2, describing some aspects of the Petersburg school of probability whose main
representatives were Chebyshev, Markov and Liapunov.
In many instances I changed the numeration of the formulas and I subdivided into sections
those lengthy papers which were presented as a single whole; in such cases I denoted the
sections by numbers in brackets, for example thus: [2]. My own comments are in curly
brackets and I often referred to my articles in the Archive for History of Exact Sciences
although I have recently generalized my contributions in a single book, History of the Theory
of Probability to the 20th century. Berlin, NG Verlag, 2004.
Almost all the translations provided below were published in microfiche collections by
Hnsel-Hohenhausen (Egelsbach; now, Frankfurt/Main) in their series Deutsche
Hochschulschriften NNo. 2514 and 2579 (1998), 2656 (1999), 2696 (2000), and 2799
(2004). The copyright to ordinary publication remained with me. Note that those collections
also contain not reprinted material.
I am rendering the terms Central limit theorem, Law of large numbers, and Method of least
squares as CLT, LLN, and MLSq, respectively, and in the lists of references I am using the
following abbreviation:
Akad. Nauk SSSR (Union Soviet Socialist Republics) = AN SU
Istoriko-Matematich. Issledovania = IMI
Imp. Akad. Nauk (or Acad. Imp. Sci.) = AN Psb
Matematich. Sbornik = MS
L, M, Psb, (R) = Leningrad, Moscow, Petersburg, in Russian.
Note: During 1914 1924 Petersburg was called Petrograd.
It is hardly amiss to add that, first, the same publisher (NG Verlag) has put out, in 2004, in
my translation,
1. A collection of my papers initially published in Russian: Russian Papers on the History
of Probability and Statistics.
2. Chebyshev, P.L. [Lectures in] The Theory of Probability.
3. Chuprov, A. Statistical Papers and Memorial Publications.
4. Nekrasov, P.A. Theory of Probability (collected papers, mostly of Nekrasov himself,
Markov and Liapunov).
Second, a few of my translations have appeared/will appear in periodicals:
1. Bernstein, S.N. (1947), Chebyshev and his influence on the development of
mathematics. Math. Scientist, vol. 26, 2001, pp. 63 73.
2. Kolmogorov, A.N. (1948), E.E. Slutsky, an obituary. Ibidem, vol. 27, 2002, pp. 67
3. Bernstein, S.N. (1912), Demonstration of the Weierstrass theorem based on the
theory of probability. Ibidem, to appear in 2004.
4. Sheynin, O. (1997), Markov and life insurance. Ibidem, to appear in 2005.
5. Khinchin, A.Ya. (1961), Mises frequentist theory, etc. Science in Context. To appear
6. Soloviev, A.D. (1997), Nekrasov and the central limit theorem. In Youshkevich:
Memorial Volume. Paris. To appear in 2005.
Acknowledgement. Dr. A.L. Dmitriev (Petersburg) sent me photostat copies of papers
NNo. 15 and 20 published in sources hardly available outside Russia.
1. Paevsky, V.V. Eulers work in demography.
In collected papers Leonard Euler. Moscow Leningrad, 1935, pp. 103 110
[1] It is known that by no means had Eulers amazing genius only manifested itself in the
field of pure mathematics. Along with his most fundamental mathematical works contained
in the surprisingly long list of his contributions, it is possible to find many writings on
mechanics, on some sections of physics (optics, etc), astronomy, exterior ballistics, on the
theory of music, the theory of some games, and on many other branches of science. Less
widely known are his works in the field of mathematical statistics, or, more correctly, of one
of its parts, mathematical demography. Nevertheless, Euler was in essence the first in this
area to lay quite a distinct mathematical foundation for a number of the main demographic
concepts such as the sequence of extinction (mortality tables), increase in population, the
period of its doubling, etc. In addition, Euler formulated, as lucid as possible, the guiding
rules for establishing the institution of private insurance (of life insurance in all its forms).
Here, I intend to touch on Eulers work in this field. It is known that, in spite of his great
fame among his contemporaries, Eulers quite exceptional scientific fertility often led to his
being unable to find publishers for his permanently appearing new works. As a result, his
writings became scattered in a great number of various editions. In particular, those devoted
to demographic problems, now collected together in his Opera omnia [1], had first appeared
in the memoirs of the Berlin, or of our Academy, and in his {previous} collected works,
whereas some of them remained unpublished and were only revealed to demographers in this
[2] His Recherches [2] and Sur la multiplication [3] deserve maximal attention from
among his contributions. In the first one Euler begins by clearly formulating the concept of
the sequence of extinction or survival that underpins the entire modern teaching of mortality
tables. Then, in a series of Questions and answers to them, he expounds the entire
methodology of applying them. Here, Euler in essence sets forth, with unusual clarity and
logical completeness, the whole main part of the mathematical (or, of the so-called formal)
theory of mortality. In addition, he introduces, for the first time ever, the new concept of
probable duration of life 1.
Along with a theoretical treatment of the measurement of mortality, Euler solves a number
of practical problems belonging to insurance mathematics. His pertinent numerical
calculations were based on the then famous Kersseboom 2 mortality table of 1742 that he
reproduces here.
In the second part of the Recherches Euler considers the increase in population. After
precisely defining the concept of increase, he solves, again by {methodologically} applying
Questions, several problems on determining this increase, on the future population, on the
number of deceased, etc. Here also he dwells on the issue of the so-called laws of mortality.
It should be noted that, whereas Euler, in the first part of his memoir, provided a quite
complete theory of formally measuring mortality at different ages, to which all the
subsequent science of demography had not added anything essentially new, he introduced
simplified assumptions on birth-rate in its second part.
Having no results of censuses (in the modern sense of this word) at hand, Euler was unable
to recognize all the importance of the problem of fertility as a function of age, or of the great
variability of the coefficients of reproduction at different ages. As a result, from our point of
view his proposition, on which he based his determination of the increase in population with
time, proved too simplified.
In his calculations, Euler supposed that the birth-rate was constant. He had not noticed
that, generally speaking, this was not true even for fertility invariable over different ages.
Indeed, the age structure of the population is determined by past events and changes
inevitably. Nevertheless, his investigation led Euler to the celebrated proposition on the
increase in population (in the absence of perturbation factors) by a geometric progression.
Even now, this principle (true, under a number of restrictive conditions) is usually taken as
the basis of pertinent hypotheses. As stated above, Euler was unable to make use of censuses,
but his great intellect prompted him to sense the importance of their data for solving
problems in measuring mortality.
In the concluding sentences of the Recherches, Euler, without using the word census,
nevertheless arrives at a quite modern idea of constructing mortality tables by means of data
on the age distribution of the living and the deceased (for a given year).
The second mathematical-statistical memoir, Sur la multiplication, is closely connected
with Eulers statistical treatment of some chapters of Pastor Sssmilchs famous book [4].
Being a son of his century, and, besides, having grown up and being reared in a pastors
family, Euler time and time again participated in argumentative defence of Christianity. It is
ascertained that he played a great part in the revision of the second edition of the Ordnung
[4], and that, apparently, he wrote to a considerable degree its Chapter 8, Von der
Geschwindigkeit der Vermehrung und von der Zeit der Verdoppelung And his memoir Sur
la multiplication (written partly in French and partly in Latin) is an expanded mathematical
version of the materials on which this chapter of Sssmilchs Ordnung was based.
Both these writings (Von der Geschwindigkeit and the French Latin memoir) contain a
number of extremely interesting and profound propositions belonging to mathematical
demography. Their main subject-matter is the determination of the time needed for a closed
(i.e., not liable to migrations) population to double under various hypotheses on mortality and
birth-rates. Without going into detail, I note that Euler had here intentionally simplified his
main assumptions so as to make their mathematical corollaries as simple and as clear as
possible. He had not considered the age composition of the population as a function of two
variables (time-epoch and time-age) as it is done nowadays, but, instead, issued for example
from the following artificially simplified pattern of the structure of the population
1) All marriages (Euler only dwelt on marital fertility) are contracted exactly at age 20 (at
the same age for man and wife).
2) Each marriage yields six living children, three boys and three girls.
3) All children are twins: a boy and a girl are born each time when their parents are exactly
22, 24 and 26 years old.
4) Nobody dies until age 40, but everyone dies at 40.
Under these conditions, beginning from the time when the husband and the wife
comprising the first pair were exactly 22 years old each, the number of births which occurred
bi-annually form the following sequence:
2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 4, 2,0, 0, 0, 0, 0, 0, 2, 6, 12, 14, 12, 6, 2,
0, 0, 0, 0, 2, 8, 20, 32, 38, 32, 20, 8, 2, 0, 0, 2, 10, 30, 60, 90, 102, 90, 60,
30, 10, 2,2, 12, 42, 100, 180, 252, 282, 252, 180, 100,42, 14,
Euler states that the terms of such sequences are the coefficients of the expansion
8achieved by division) of some generating function and indicates the form of this function
for another numerical example [3, p. 551]. He [4, Chapter 8, 161] also notes that, in spite of
its extreme irregularity which becomes evident when considering only the first terms, the
sequence, when continued sufficiently far, begins to transform itself into a geometric
progression, and all its irregularity disappears:
However irregular these sequences appear initially, after being ever
continued they will nevertheless finally turn into geometric progressions,
and the irregularities noticed at the beginning will decrease the more, the
further, until finally disappearing almost completely.
About 160 years later, the German {German American} statistician and mathematician
Gumbel [5] proved Eulers statement for the abovementioned sequence by indicating its
generating function, 2/(1 x11 x12 x13). He also showed that the common ratio of its
limiting geometric progression was the greatest in absolute value root, x = 1.0961 3, of the
appropriate equation,
x13 x2 1 = 0.
In each of his writings, under the most various hypotheses on birth-rates and mortality,
Euler invariably concluded that for a closed population the number of births (and therefore
the population itself) grows by a geometric progression. Of course, modern demographic
science cannot entirely agree with such a statement. The growth of a given population
depends on so many factors, most of them unyielding to interpretation and being of a social
nature, that it is hardly possible to hope to represent it by a mathematical formula, let alone
by such a simple expression. However, modern mathematical demography does not decline
to study additionally abstract patterns of the changes in the population and its structure,
although by no means attaching to them the rank of predictions and considering them only as
an aid for determining the most perfect indicators of the changes observed in populations.
And the patterns themselves are now remote from those primitive and simple propositions
which Euler had to accept as his basis. This applies above all to the concept of the age
structure of the population that modern demographers got used to consider as continuously
changeable in time-epoch.
It is interesting, however, that, assuming that some demographic factors (as mortality and
fertility at different ages) are constant, the most subtle methods of constructing abstract
patterns of the changes in population largely lead to the same Eulerian conclusions about its
limiting growth by geometric progressions Such, for example, was the result attained by the
most eminent American demographer Lotka who made use of integral equations of the
Volterra type, see [7; 8], or by the German researchers Bonz & Hilburg [9] who determined
the growth as the solution of a difference equation.
[3] In addition to Eulers purely demographic works, as they might be termed, see above,
he left a number of memoirs devoted to practical issues of insurance:
1. On life annuities [10]. After defining the main notions (i.e., mortality and discount) that
form the basis of any insurance operation, Euler offers recurrence formulas determining the
current cost of life (postnumerando) annuities and provides a number of practical tables
which he constructed on the basis of the Kersseboom mortality table.
2. On widow funds [11]. Here, Euler considers problems connected with insurance
depending on the survival or death of two persons (on two lives), and with life insurance in
general. He comes close to the concept of commutative numbers later introduced into the
science of insurance by Tetens and very important for modern actuaries.
3. On public institutions for widows and in case of death [12]. This is Eulers most
extensive writing on insurance mathematics. It consists of three parts
a) On widow pensions.
b) On mutual insurance against death; and
c) On the plan of a new tontine (which provided a possibility of new members being
admitted continuously).
This is a vast treatise on insurance combining much written by Euler previously and supplied
with a large number of tables not devoid of interest even nowadays.
4. On insuring orphans [13]. Here, Euler determines the cost of insurance to be received
by the heirs upon the death of both their parents. Once more making use of the Kersseboom
mortality table, he solves a number of separate pertinent problems by means of one general
equation.
5. A memoir (more precisely, a draft fragment first published in the Opera omnia) on
calculating tontine annuities [14]. Here. Euler considers various modifications of tontines.
The fragment apparently represented an examination of a tontine proposal formulated by him
on the request of Friedrich II.
[4] uler had always been interested in problems of mortality, birth-rates and growth of
population; he repeatedly turned to them in ad hoc memoirs and they also served him as
examples in his purely mathematical writings. Thus, in Chapter 6 of his Introductio [15]
Euler supplied four demographic (although extremely simple) problems as illustrations for
calculations. The following one is characteristic for his time:
After the Flood, mankind was reduced to six people. Supposing that in 200
years the population had increased up to 1 000 000, determine the yearly
rate of its growth.
It might be thought that, being a remarkable representative of a purely mathematical
genius, Euler will restrict his efforts with a formal mathematical treatment of his problems.
Actually, however, the situation was much different. His profound intellect compelled him to
penetrate into the very essence of the problems under his study. He understood perfectly well
the relation between his patterns and the facts of concrete reality and made a number of
extremely interesting remarks on this subject. Thus, concerning mortality tables he warned
against the desire to believe that the determined order of extinction was universal. He noted
that each table was only applicable to that city, or province, for which it was constructed. He
also stated that mortality in towns must be lower than in the large cities, but higher than in
the villages, and this proposition was apparently true for his epoch.
Euler then remarked that the order of extinction calculated by means of mortality tables
ceased to be applicable in any cases of epidemics, famine or war. He understood perfectly
well the importance of migratory factors in constructing mortality tables, correctly allowed
for the difference in mortality of men and women, and repeatedly indicated that the
application of mortality tables constructed for both sexes taken together could lead to large
errors, he thus criticized the Kersseboom table.
Not only did Euler, time and time again, return to the problem of the comparatively lower
mortality of women, but, what is especially remarkable, he had a perfect understanding of
purely demographic problems of mortality in early childhood (during the first months of life).
On this subject he formulated well-thought out ideas often escaping the notice even of
modern specialists in public hygiene when they compare the mortality of all children and of
those constituting isolated groups (in nursery schools and crches).
Euler noted that a mortality table based on observing life annuitants did not express the
actual mortality of the general population, the less so for infants younger than one year. The
observations, as he stated, undoubtedly only covered children who had escaped the dangers
of the first months of life, so that the table represented the mortality of a selected collective of
children rather than of the entire children population. Indeed [2, pp. 86 87]:
In this respect one should not apply the records of the life annuities which
begin with infants older than a year. Because, first of all, these infants
cannot be considered as newly born, and most of them have undoubtedly
escaped the dangers of the first months; and, further, infants of feeble frame
often are not included at all, so that one should regard the infants, for
whom life annuities were taken out, as selected
And, considering the numbers of survivors according to the Kersseboom table, Euler
Therefore, because the table is constructed for selected infants who had already survived
several months after birth when wishing to apply it to all the newly born infants of a city or
province, all its numbers should be decreased by a certain part to allow for the high
mortality, to which the infants are at once subjected after being born.
How remote are these considerations from dry and formal mathematical patterns! Euler,
the mathematical genius, shows here his features of a real demographer, deeply penetrating
into the concrete essence of the problem under study.
Naturally, not everything created by Euler in the field of demography retains absolute
importance for our time. Something became obsolete and is only of a historic interest.
However, he by right occupies a distinguished place in the cohort of remarkable
mathematicians who participated in studying demographic and related problems, in the
cohort that had begun with De Moivre and included himself, Fourier and Laplace, and to
which Gauss was not alien either; to which Volterra, von Mises, Frchet and many others
have recently joined.
1. {Huygens effectively applied this notion in his correspondence of 1669, see my paper in
the Arch.Hist. Ex. Sci., vol. 17, 1977, pp. 247 248.
2. The Dutch scientist Wiliem Kersseboom (1691 1771) held financial posts and
constructed his mortality table, common for both sexes, by basing his calculations on
observations of the longevity of persons who had bought annuities for themselves from the
3. Gumbel omitted the numerical factor (2) in the generating function [6,p. XL].
1. Euler, L. (1923), Opera omnia, ser. 1, t. 7. Leipzig Berlin.
2. --- (1767), Recherches gnrales sur la mortalit et la multiplication du genre humain [1,
pp. 79 100].
3. --- (Manuscript), Sur la multiplication du genre humain [1, pp. 545 552].
4. Sssmilch, J.P. (1761 1762), Die Gttliche Ordnung in den Vernderungen des
menschlichen Geschlechts, etc,Bde 1 2. Second edition. Gttingen Augsburg, 1988.
5. Gumbel, E.J. (1916), Eine Darstellung statistischer Reihen durch Euler. Jahresber.
Deutsch. Mathematikervereinigung, Bd. 25, pp. 251 264.
6. Du Pasquier, L.G. (1923), Prface to [1].
7. Dublin, L.I., Lotka, A.J. (1925), On the true rate of natural increase. J. Amer. Stat.
Assoc., new ser., vol. 20, pp. 305 339.
8. Sharpe, F.R., Lotka, A.J. (1911), A problem in age distribution. London, Edinb. and
Dublin Phil. Mag., 6th ser., vol. 21, p. 435ff.
9. Bonz, F., Hilburg, F. (1931), Die voraussichtliche Bevlkerungsentwicklung in Deutschland. Z. angew. Math. Mech., Bd. 11, pp. 237 243.
10. Euler, L. (1767), Sur les rentes viagres [1, pp. 101 112].
11. --- (1770), Nthige Berechnung zur Einrichtung einer Witwencasse [1, pp. 153 161].
12. Fuss, N., sous la direction de Euler (1776), Eclaircissements sur les tablissements
publics en faveur tant des veuves que les morts, avec la description dune nouvelle espce de
Tontine aussi favorable au Public quutile lEtat [1, pp. 181 245].
13. Euler, L. (1785), Solution questionis ad calculum probabilitatis : Quantum duo
persolvere debeant, etc. [1, pp. 393 407].
14. --- (Manuscript), Sur le calcul des rentes tontinires [1, pp. 553 577].
15. --- (1748), Introductio in analysin infinitorum, t. 1. German translation: Berlin, 1885.
2. Daniel Bernoulli. On the Mean Duration of Marriages for Any Age of the Spouses
and on Other Related Issues
Translated from Russian translation in Ptukha, M.V.
(History of Statistics), vol. 1. Moscow, 1955, pp. 453 464...
Translators Foreword
I have translated Bernoullis memoir [3] from its Russian translation (see Contents)
retaining the several Latin phrases that were left in its Russian text. With respect to the
mathematical background needed here, Bernoulli referred to his previous work [2] calling it
preliminary (16). There, he attacked the same issue formulating it, however, as an urn
problem and solved it by two methods, by a combinatorial approach and by applying
differential equations [4]. For that matter, he was the first to use such equations in probability
in his memoir of 1738 generally known owing to the introduction there of moral expectation.
I have not fully reproduced the tables; readers can find them in the original Latin version
of the memoir.
1. Since a perpetual substitution [commutatio] of a person who has to die by one born is
going on, very many years ago various nations had compiled tables both of those born, and of
the deceased, by whose means the laws of variations and reciprocities were observed and
finally established with a surprising success. Because, although the fate of each separate man
is utterly unknown, it cannot nevertheless be denied that, for a large arbitrarily taken number
of things, the mean state corresponds to nearly invariable laws regardless of what this state
concerns or of the issue at hand whatsoever.
Thus, it was observed that the number of the yearly born boys was invariably greater than
that of the yearly born girls, and, what was more surprising, in each country the inequality
itself between these numbers almost invariably retained one and the same proportion with
respect to the total number of newly born. However, this last {property} can only be noticed
for very large numbers in which the random fate of one or another phenomenon occurs to be
hardly noticeable if only bearing in mind its ratio to the total quantity. Even these very
proportions for one and for the other sex little differ from each other in different countries.
The most thoroughly compiled tables indicate, among other facts, the causes of death
wherefrom we perfectly well acquaint ourselves with the distinctive peculiarities of deadly
diseases. It is known now that in our days smallpox alone destroys a twelfth or a thirteenth
part of each generation either more or less depending on the way of life and the differences
between the nations. It is also known that during the first year of life diseases of the first
childrens age carry off almost three tenth of the entire generation.
Then, what is closer to our goal, it was observed that in general women enjoy a longer life
than men; regarding this, we have a table compiled by the illustrious Wargentin in Sweden
which quite confirms that observation. And this last-mentioned circumstance cannot be
attributed to the different ways of life led by men and women because the womens
mentioned privilege most invariably reveals itself from the very first nappies and holds good
with them during all their life. 1623 boys and 1438 girls die during the first year of life from
among one and the same number of persons of each sex. The mean duration of life taken
from the very day of birth is 24 years and 2 months for boys and 28 years and 10 months for
girls; 2008 souls of the male sex live to the twentieth year and 2337 of the female sex. Hence,
it is seen that the number of living women invariably exceeds the number of living men.
Whether we consider such observations from the point of view of politics, or medicine,
they will not lose their importance; on the contrary, if the authors will understand all their
value and properly use them, their application will become even wider.
2.In addition to tables of the born and the deceased, tables of marriages are also usually
compiled, which, above all, serve for illuminating government economy, and will serve this
purpose still better if compiled more thoroughly. It would be desirable to designate in the
records by additional Arabic numerals the age of each of the marrying persons, and, by
another Roman numeral, what marriage does the bridegroom enter, and what marriage the
bride, the first, the second, or the third one. From tables of marriages compiled in this
manner the customs governing this matter in different nations and different towns become
very intelligible. A sensible attack against these customs, if they do not quite correspond to
state intentions, could be launched.
3. The mentioned thorough tables can only be compiled on the basis of permanent
observations collected with great care. The use of prior considerations does not help here at
all, but how many new truths would it be possible to elicit therefrom by theoretical reasoning
alone! Mortality tables, for example, show us the mean duration of the remaining life for
persons of any age; true, they show it not directly but through deductions which somehow all
by themselves can randomly cross the mind of anyone. However, the problem with which we
shall now deal, the one concerning the mean duration of marriages, is much more difficult
and demands a lot of deduction.
Since, on the other hand, it affects various groups of mankind, the rights of succession and
mutual relations, I think that I shall do a valuable deed if I point out the way for ascertaining
a subject which has much in common with an essay promulgated by me seven years ago in
the Paris Acadmie Royale des Sciences and inserted in the Commentaries {Mmoires} of
that Academy for 1760 [1]. There, applying a kind of analysis unusual for that matter, I
spoke in detail about mortality from smallpox.
4. Since the problem concerns the mean duration of marriages for a given age of the
spouses, it is necessary first of all to compile a table for some number of homogeneous
marriages that will show how many such marriages remains untouched by death on the
termination of each subsequent year of life. Nevertheless, a method by whose means such a
table could have been compiled, is not yet discovered. True, mortality tables show the
numbers of both men and women still living after each year of their life. However, either
number consists of the widowed and of those married, and because each of these two
numbers is absolutely unknown, the application of mathematical analysis, by whose means
these numbers could have been calculated, is required. This analysis should be borrowed
from the theory of probability.
I shall begin with a simpler issue. First of all we assume the same age for both sexes, i.e.,
that all the marrying bridegrooms and brides are of one and the same age. Let us suppose
further that men and women of the same age are in an equal measure running the danger of
death although observations show that the case is somewhat different and that the weaker sex
is leading a way of life safer from death. The latter proposition, however, should be
understood not as though a smaller number of women die yearly in populous cities or other
places than men do. If as many women were born as men, then as many of them would have
been dying. However, as it is known, more women are living in any nation than men
wherefrom it follows that the yearly numbers of those dying of both sexes can even be
absolutely equal. But this does not mean that, with respect to the total number of the living of
both sexes, women do not die in a lower proportion than men. I wanted to recall this point
because I noted that some authors express a wrong opinion about it.
Thus, after examining considerations in favor of our general hypothesis, I shall show in
how it can be verified altogether and with utmost precision.
5. Suppose that the initial number of all the marriages = n so that the number of all those
married will be 2n. Then let their part carried off by death after a known number of years be
counted. The number of those still living we denote by r so that the number of all the
deceased will be (2n r). We denote the number of the remaining marriages by x, then the
number of all the widowed will be equal to (r 2x). Having determined all this, I say that x =
(rr r)/(4n 2).
I provided an obvious solution of this problem in the memoir [2, 2]. True, I have there
applied terms usually accepted in the theory of probability, but everyone can see that exactly
this is the essence of the matter.
Then, the number of all the widowed, or (r 2x), will be (2nr rr)/(2n 1) whose half
thus expresses the number of either the widowers or the widows because we suppose that
death equally strikes down men and women.
6. Consequently, it is now easy to determine the number of marriages which will remain
untouched by death from among any number of those still living. Since this last number is
given in the mortality tables for each year of the age, the number of remaining marriages can
be determined at the same time for each year. I select the mortality table for the city of
Breslau compiled by the most illustrious Halley and reprinted in various works. In particular,
it can be found in the excellent work of Deparcieux entitled Essai sur les probabilits de la
dure de la vie humaine {1746}. I shall use it for compiling new tables suitable for our goal.
I may naturally begin with any number of marriages and any common age. Suppose that
there are 500 initial marriages contracted between 1000 persons each of them having the
exact age of 20 years. Since, in accord with the Halley mortality table, 598 persons {out of
1000} reach twenty years, whereas I assume 1000 persons, each of Halleys numbers ought
to be increased in conformity with the proportion between the numbers 598 and 1000 {1000
and 598}. I discard the fractions and substitute instead of them integers to which they are
closer or which can better reveal the uniformity in the movement ahead [qui melius
uniformitatem in progressu observat].
The table appended below consists of four columns. The first one shows the age expressed
in years; the second column is the number of persons still living; the third one is the number
of remaining marriages not destroyed by death; the fourth column shows the number of the
widowed without distinguishing between their sexes. These last numbers should be divided
into two equal groups, those of widowers and widows; all those who became single after
their first marriage {was broken}, irrespective of whether they contracted new marriages or
remained widowed, should be entered into this column.
Table to 6
Age in Years Remained Living Remained Marriages Widowed
Note: Daniel Bernoulli compiled this table up to and including age 89.
7. The formula that we made use of for finding the values in the third column by
calculation, to wit, (rr r)/(4n 2), clearly shows that its numbers are not directly
proportional to the first arbitrarily taken number if only this initial number is not very large.
However, even then the subsequent numbers will ever more move away from the unity of
proportion. Then, it should be noted that since we have discarded all the fractions, small
mistakes larger than 1 and less than 2 could have crept into the numbers of the fourth
column. In general, these numbers correspond to the formula (2nr rr)/(2n 1) (5).
Consequently, the number of the widowed reaches its maximum when the initial number of
those marrying decreases to its half, i.e., when x = n. In this case the maximal number of the
still living widowed will be equal, in accord with the formula, to nn/(2n 1), or, in our
example, 250 , although the table shows 252. I recall this so that someone will not attribute
the committed small mistakes, made so as to facilitate calculations, to our method itself.
8. The table that we just adduced begins with 20 years. It is not difficult to compile a new
table beginning with any other age. Suppose that it is desired to study the peculiarities
[indolem] of the marriages for the age of the bridegroom and bride who are exactly 30 years
old. The previous table provides the number of marriages remaining at that age as being
equal to 395. Since the number of those still living is always equal to twice the number of the
marriages, the second column should begin with 790 persons. Consequently, the number of
those living, or 888, should be replaced by 790, and all the subsequent numbers given in the
second column should be decreased in the same proportion. As to the numbers in the third
column, all of them should be the same as they were previously. Finally, if the doubled
numbers of the third column be subtracted from the numbers of the second column decreased
in the indicated way, the numbers of the fourth column will be obtained. Tables for any
common (for men and women) age of those contracting marriages are compiled in the same
9. If it is further asked, when will the initial marriages contracted at the age of 20 decrease
by one half owing to deaths, this is clearly seen at once when looking at the table. Since 500
is assumed as the initial number of marriages, we should only look in the third column at
what age 250 marriages remain intact. Then we shall indeed see that this number corresponds
to the age between 42 and 43 years of life of the spouses, or, more precisely, to 42 years and
4 months. Consequently, we can bet, with equal risks, on whether or not a marriage,
contracted at the age of 20 years common for both spouses, will exist after 22 years and 4
months. The same question for any other age of those marrying, if only this age is the same
for both spouses, is solved in exactly the same way. Let for example the initial age of the
spouses be 40 years. Our table shows that at this age 276 of the initial marriages remain in
existence and that this number is decreased by one half after 12 years and 10 months.
In this way I compiled the appended short table for the beginning and the end of each fiveyear period. The first column shows the common age of both spouses; the second one
indicates the time expressed in years and months after which a half of the marriages will
probably be destroyed. By the method of interpolation these quantities can be determined
rather precisely for any intermediate age.
{Two tables combined} Table to 9 and 11
Age in Years
Duration of Marriages, Years & Months
Note: Daniel Bernoulli compiled both his tables with interval 5 years.
10. The question about the time when half the marriages will be destroyed at whatever age
they began, should not be confused with our fundamental question on the mean duration of
marriages expected at any age of those marrying if only it is the same for both spouses.
However, it is easy to foresee that these questions will not very considerably differ from each
other. The method for finding the mean duration of marriages resembles that which we apply
for determining the remaining duration of life expected at some age. But this method requires
that the values of each yearly wastage be known. Since such losses were not yet determined
by any observations, it crossed my mind whether this can be done by calculations with
respect to any age, and I have indeed accomplished this work in accord with my desire [ex
sententia]. And so, I shall show now how, by means of the table appended to 6, the
remaining duration of marriages for any age of the spouses should be looked for.
11. It is necessary to add together all the numbers indicated in the third column of our table
beginning with the given age to the end, and to divide the sum by the number concerning the
given age. The quotient obtained will show, as it is known, the sought mean duration even if
the separate marriages from among those destroyed during each given year were actually
only destroyed at the end of the year [simul in fine anni]. Indeed, since the destruction is
going on almost uniformly during the whole year, there will not be a perceptible mistake in
attributing the obtained sum to the mean yearly number of {the remaining} marriages.
It follows therefrom that the said quotient should be decreased by half a year, or by six
months. Thus, if, for example, the matter will concern the mean duration of marriages
between spouses 55 years of age, the sum of the numbers of the third column should be taken
beginning with 118 and to the end; this sum equals 1208. It should be divided by 118, by
the number corresponding to the initial age. The quotient will approximately be 10
designating the same number of years. Subtracting half a year from this, we obtain 9 or
nine years and nine months. Consequently, the mean duration of marriages between 55-yearold spouses equals nine years and the same number of months. []
12. So that it would somehow become clear how much the results of our theoretical
computations fit in with observations; I choose one example that I found recently in the Bern
Acts 1 where it is said that 1053 remaining marriages are recorded in Lausanne and that 49
new marriages are contracted there yearly. It follows therefrom that the mean duration of
these recently contracted marriages is 21 years and six months. Such a mean duration of
marriages is longer than that which corresponds to the age of 25 years in the table. However,
it is not at all likely that the mean age was about 25 years for each of the contracted
marriages so that our calculations quite well fit in with observations. At present, I have no
time [non vacat] for looking up other examples offered by other authors. If others will desire
to take upon themselves this matter, it would be easy for them to draw up their own opinion
about it, let them only remember that mortality in different countries is only somewhat
[paululum] different and that depending on the per cent of mortality the duration of marriages
increases or decreases.
13. Now, if we desire to find out the total number of all the existing marriages in a country
where 500 new marriages are contracted yearly, if these, according to our supposition are
only contracted between persons having 20 years of age, then, to this end, the mean duration
of marriages should be multiplied by their yearly number, i.e., 23 5/6 multiplied by 500. The
obtained product, or 11 917, will provide the required sum: it will be the number of all the
existing marriages. And the total sum of all the people living in conjugal union is 23 834. If
we take some other initial age for persons contracting a marriage, then the table of 11 can
help {to solve} this problem. The number of all the preserved marriages is everywhere equal
to the product of the marriages contracted during one year by the mean duration which is
recorded in the second column of this short table.
Suppose that the conclusion of each marriage is attributed to the end of the 30th year [in
finem trigesimi anni]. Then, it will be necessary to multiply the number of the yearly
marriages by 18 5/6. The number of yearly marriages itself will now be less because it
depends on the number of those still living; that is, on the numbers 1000 and 888 which
correspond to the ages of 20 and 30 years. Consequently, the number of yearly marriages will
be equal to (888/1000) 500, or 444, which, being multiplied by 18 5/6, gives 8362 whereas it
was 11 917 for the age of 20 years. It can be said already in advance that the marriages of the
first kind will be more fruitful than those only contracted after the end of the 30th year [finito
trigesimo anni] of age.
If some state will gain 2300 children yearly from 500 yearly marriages each of them
contracted before the end [sub finem] of the age of 20, then this same state will be hardly
capable of obtaining more than 1000 children if all the marriages are, and each of them
separately is contracted only after the lapse [elapso trigesimo anno] of the 30th year of age 2.
14. I shall add some words about the number of all the widowed in the same state about
which we spoke just now; suppose that they invariably remain widowed or that all men and
women once becoming widowed are attributed to this category. If, in the table of 6, the
values of the fourth column are added together, the number 10 326 will be obtained which is
the approximate expression for the number of all the persons widowed until the end [usque
ad finem] of age 89. About 24 souls outliving this age may be added here so that the number
of all the widowed in that state may be reckoned as 10 350. Consequently, all the people
connected into a conjugal union will be to all the widowed as approximately 23 834 to 10
350 or almost as 23:10. The matter is going on in such a way if we suppose that each
marriage is contracted at the age of 20 for both spouses. Another age will have another table
with other numbers which can be easily obtained after appropriate changes in the
calculations. It would be too bold to speak about everything. There are very many issues that
are resolved by calculations alone and certainly better than by the most diligent observations
[et quidem melius guam observationitas utcunque assiduis].
15. I pass on to another kind of marriages contracted between spouses of differing ages.
The common feature for marriages of this kind is that the husbands are mostly older than the
wives. It is known, however, that the danger of death heightens with age, and, besides, that
under equal conditions men are subjected to greater dangers than women. Thus, if we assume
a higher age for the husbands, then they will both easier and sooner [facilius et citius] pass
away and leave behind them far more widows than there will be widowers. True, I thought
that this inequality between the numbers of widowers and widows will present too much
difficulties for our subject. However, the very nature of things helped in that a differential
equality {equation} between three unknown quantities fortunately admits not only
differentiation but integration as well. I applied such an approach in the memoir [1]
mentioned above in 3. And so, it is understandable that I made use of the method of the
algorithm of infinitesimals which requires that the most greater number, or as though an
infinite number be taken as the initial number of marriages. I have expounded this method in
full with only a change of names in a preliminary essay [2, from 7 to the end]. Here, we
have husbands and wives, there, we had black and white cards; here, marriages, there,
pairs of cards; here, finally, a higher mortality of one of the two sexes, there, a greater
facility for being drawn from an urn, or a greater inclination for leaving it.
16. Let us again assume n as the initial number of marriages. Suppose that each marriage is
contracted in such a way that all the husbands are of the same age one with another as are all
the wives, but that for either sex the ages are different. Then, let the number of the still living
husbands after a known number of years be s, and the number of living wives be t; both these
numbers are determined by mortality tables. It is asked now, how many remaining or
untouched by death marriages will there be between all these living people; and, further, how
many will there be widowers, or at least those who once became a widower; and how many
widows or women who once became a widow. Again, assume that the number of the
remaining marriages is x. then the number of the remaining widowers will be (s x), and the
number of the remaining widows, (t x). Under all these assumptions, as we proved in 7 of
our preliminary memoir [2], we find that x is equal to st/n. And so, if we find the numbers s
and t, all the rest will become known as well.
It is apparently not necessary for a direct ratio to exist between the different mortalities of
men and women caused by the difference in age. This, however, is clear to anyone on
account of a certain interrelation between the numbers s and t which necessarily includes
different mortalities being also presumed. For all its simplicity, our formula is so general that
it can express at the same time another distinction between the measures of mortality caused
by the difference between the sexes if mortality tables for men and women separately will be
Then, to be able to use the Halley tables, we shall only consider the difference between the
measures of mortality caused by the difference in ages. We did the same when speaking
about the marriages between spouses of equal ages. It will be thus possible the better to
compare the various kinds of marriages with each other.
17. I consider it needless to say how another table similar to the one appended to 6 can be
compiled when the ages of the spouses are unequal. True, any difference between the ages
requires new calculations, but it is sufficient to provide the data for five-year periods because
the intermediate values will be rather precisely reckoned by interpolation. The example that I
shall now present will be a specimen for all cases.
Suppose that new marriages are contracted between 500 girls each of them being 20 years
old and the same number of men, 40 years old each. Thus, the number of marriages is again
500, but now the numbers s and t for each year, i.e., the numbers of both men and women
still living, have to be found. Both numbers can be precisely and, besides, easily determined
through calculations by means of the second column of our table appended to 6. Indeed, the
number t will be obtained when taking a half of each number indicated in the second column;
if this half is then multiplied by 1000/744, we obtain the number s. This multiplication is
easy for the reason that the initial number of men is assumed to be 500. In that way the
second column is compiled; it provides the number of still living men and women after each
year of life. Then the third column is compiled. It shows the number of the remaining
marriages by assuming, for each year, the magnitude st/n. Finally, the fourth and the fifth
columns showing the number of widowers and widows are compiled by subtracting the
number of the remaining marriages from the number of the still living men or women. The
calculation of the last-mentioned numbers is of no independent significance. I considered it
necessary to adduce them by issuing from the other columns of the table.
Table to 17
Age in Years Remained Living Remained Widowed Widows
Men Women Men Women Marriages
Note: Daniel Bernoulli compiled this table up to and including
ages 99/79 of husbands/wives
18. The use of this second table is the same as of the previous one. Consequently, all about
which we spoke in 7 14 might also be applied for the present {the second} kind of
marriages. Thus, for example, when in 9 I formulated the problem about determining the
time when half the marriages contracted between persons of the age of 20 years will be
destroyed, I found that this will happen after 22 years and 4 months. For the marriages of
the present kind, contracted between women having 20, and men of 40 years of age, this time
is decreased to 16 years and 4 months as is seen just by considering the third column of the
second table. Had a similar problem about marriages between women of 40, and men of 60
years of age be raised, the same third column would have shown that 202 marriages will
remain and that they {that this number} will decrease to 101 in 9 years and one month.
Thus, it offers no difficulty to compile a short table similar to the one appended to 9, and
it only suffices to remind about this. However, since the matter mainly concerns the mean
duration of marriages for differing ages of the spouses, I do not shirk this work either.
19. This problem, when the husbands are 20 years older than the wives, is solved by an
absolutely the same method which we made use of in 11where we assumed that the age of
both spouses was the same. And so, for the initial age of men being 40 years and of women,
20 years, it will be necessary to take the sum of the numbers of the third column in the first
table, which is equal to 9207, and to divide it by the number of the initial marriages, or by
500. The quotient will be 18 41/100 years, or roughly 18 years and 5 months. Subtracting 6
months from this we shall obtain 17 years and 11 months, the sought duration of a marriage.
I have indeed compiled the appended table by this method. It can now be compared with the
short table appended to 11 so that it would be seen how different are the two kinds of
marriages with respect to their duration expected for persons of some age. The difference is
not insignificant, and it would have been still greater had we possessed statistical data about
the higher mortality of men and had at our disposal mortality tables where people of either
sex were separated from each other with the numbers of men and women shown separately.
Mean Duration of Marriages, Years & Months
Note: Daniel Bernoulli compiled this table with interval 5 years.
20. Finally, some will desire to compare both kinds of marriages, i.e., on the one hand,
marriages where both spouses are of the same age, and, on the other hand, of those where the
husband is 20 years older than the wife, so that even those who quail before new calculations
were capable of forming, without a large error, a judgement about any mean state.
a) We have already spoken about the different mean durations of marriages of both kinds.
Thus, for example, not without reason a woman 55 years old marrying a man of the same age
promises herself a twice greater duration of her marriage than if she would have married a
75-year-old man.
b) If we imagine ourselves two nations the first of them having 500 yearly marriages of the
first kind, and the second one, the same number of marriages of the second kind, then, under
the assumption that each of the two nations will remain in invariable conditions, the first state
will have {in all} 11 917 marriages and the second one, only 8958 (13). We consider
marriages contracted for the first time by either spouse and in any case such marriages make
up the greatest number {the greatest part}.
c) Then, one and the same state, where 500 marriages of the first kind can be contracted
yearly, will be unable to have more than 372 marriages of the second kind because certainly
from among 500 young men only 372 will reach 40 years. Thus, a quarter of the girls will
have to remain unmarried or marry widowers. This is why much more widowers than
widows get married although the number of the former is smaller than that of the latter.
d) Let us consider now those widowed of either sex. To these we attribute all those who
once became widowed irrespective of whether they remain widowed or contract a second
marriage. We saw in 14 that for 500 yearly marriages of the first kind the number of all the
widowed is, in accord with the indicated reckoning, 10 350, or 5175 widowers and 5175
widows. But the matter is far different when it concerns marriages of the second kind
because then the number of widowers is considerably less than the number of widows. Then
our table in 17 shows only 2154 widowers whereas the sum of all the numbers in the fifth
column is 7949, and this number of widows should still be increased by 197, by their
number yet living after the termination of the last year in the table so that the total number of
widows is 8146. However, the sum of all the widowed is almost the same in both cases.
When comparing the number of widows with the number of widowers, we see that in the
second case (where the bridegroom is 40, and the bride is 20 years old) the number of
widows is almost four times greater than the number of widowers whereas in the first case (in
which the spouses are of the same age) these numbers are equal to each other. Consequently,
in a state where 500 marriages of the first kind are yearly contracted, there will be 11 917
husbands and as many wives, 5175 widowers and as many widows, the total number of all of
them being 34 184. If, however, 500 marriages of the second kind will be contracted in that
state, there will be 9207 husbands and as many wives, 2154 widowers and 8146 widows with
the total number of them being 28 714.
e) In a state, which, let us suppose, acquired stable conditions, if 500 marriages of the first
kind are contracted yearly, the same number of them will be destroyed during the same time.
250 husbands will die leaving as many widows and 250 wives as well who will make
widowers out of 250 husbands. However, from the 500 marriages of the second kind, death
will yearly carry off 345 husbands and 155 wives; and there will thus remain 155 widowers
and 345 widows. Since the number of the remaining husbands is s, the number of the
remaining wives, t, and the number of the remaining marriages, st/n (16), the wastage of
husbands will be expressed by the ratio of the number of the former to the number of the
latter, that is, of s to st/n, or n to t. It follows therefrom that the wastage of husbands is
expressed by the formula tds/n, and the wastage of wives is sdt/n. Therefore, the total
numbers of the deceased husbands and wives are, respectively,
tds/n and
sdt/n.
However, since the general ratio between s and t is only given by mortality tables, the
integrals above can only be partly [per partes] given. I have obtained the numbers just
adduced by this method.
My essay, whatever it is, makes it clear that many variations occur with mankind and that
there are many reciprocities that may be more thoroughly and better determined by
calculation than by countless observations made until now.
1. {I can only say that the Acta Helvetica had been published in Basel.}
2. {The Russian text says Only before the lapse, etc.}
1. Bernoulli, D. (1766), Essai dune nouvelle analyse de la mortalit cause par la petite
vrole, et des avantages de linoculation pour la prvenir. Werke, Bd. 2. Hrsg. B.L. van der
Waerden. Basel, 1982, pp. 235 267.
2. --- (1768), De usu algorithmi infinitesimalis in arte coniectandi specimen. Ibidem, pp.
276 287.
3. --- (1768), De duratione media matrimoniorum, pro quacunque conjugum aetate,
aliisque quaestionibus affinibus. Ibidem, pp. 290 303.
4. Sheynin, O. (1977), Daniel Bernoullis work on probability. In Studies in History of
Statistics and Probability, vol. 2. Editors, Sir Maurice Kendall & R.L. Plackett. London,
1977, pp. 105 132.
3. Kh.O. Ondar. On the Works of Davidov in Probability Theory
and on His Methodological Views
Istoria i Metodologia Estesvennykh Nauk, vol. 11, 1971, pp. 98 109
Avgust Yulievich Davidov (1823 1885) was Professor at Moscow University from 1853
and author of studies in mechanics and of various mathematical problems. His research in
probability bordered on the work of Laplace and Poisson, and he also paid much attention to
the application of statistics in medicine. In Russia, Davidov is not forgotten, but the only
study of his findings is the one that I am presenting now in translation. Note that I had not
checked Ondars formulas and am not satisfied with his mathematical reasoning. I have left
out some parts of the text (repetitions, hardly important details, discussion of Davidovs
philosophical outlook); in Note 1 I omitted the titles of the mentioned books from Davidovs
private library.
I myself, some 25 or 30 years ago, attempted to get hold of ref. [6] but only found an
unreadable microfilm of some of its pages, and I notice that Ondar had passed it over in
Davidov published ten contributions on the theory of probability [1 10]. Modern
literature on the history of mathematics [11 15] cites only some of them, and, for that
matter, only in passing. Usually only five of his works are mentioned whereas the other ones
are not even included in the listing of his writings on probability. However, these are of
certain interest as providing a more comprehensive characteristic of Davidovs efforts in this
The Department of rare books and manuscripts of the Gorky Library at Moscow
University is keeping Davidovs private collection that includes many treatises on probability
as well as a large number of separate articles on this subject. He left many marginal notes in
these sources and it is possible to assume that he knew and studied the contemporaneous
materials on probability. At least as far as this branch of mathematics is concerned, we
hesitate therefore to agree with Vygodsky [12, p. 153], that Davidov did not like to bother
himself with reading [scientific] literature.
[1] We turn now to consider Davidovs works in probability and we begin with
contribution [4] since it is the most valuable of all of them; as is witnessed by [1], Davidov
expounded its contents to students of Moscow University. The study of [4] will allow us to
judge, to a certain extent, his scientific standards and his style, and to see the findings
contained there.
In his Introduction, Davidov indicates that conclusions, arrived at in statistics without
taking into account stochastic laws, are not trustworthy. He regrets that the theory of
probability did not become generally applied because of its involved mathematical tools, and
he aims at deriving a formula for usage in many statistical problems.
We shall now describe his further reasoning almost verbatim, since, in our opinion, he is
making use of the concepts of distribution function and density 2. Suppose that some event
dependent on n magnitudes
takes place when they satisfy a certain condition, for example when
F( 1,
n)
= 0.
Let us consider an event {stochastically}determined by a more general condition
a F( 1,
Assuming that F(x) is the probability
P(0 < < x) = F(x),
we shall determine the probability of the existence of (2). Davidov indicates that
F(x + dx)
F(x) + F (x)dx
P(0 < < x + dx)
F(x) + F (x)dx.
F (x)dx = f(x)dx,
P( = x)
f(x)dx
P(m < x < n) =
f( )d .
Let now
i;
i = 1, 2, , n; then
fi( i)d i = 1.
Of course [4, p. 3], the s are supposed to be continuous.
Had Davidov taken the lower boundary of equal to rather than to zero, expression (3)
would have been the modern definition of distribution function, and (4) would have
corresponded to the definition of density. []
We now take up Davidovs further deliberations. Suppose that in inequalities (2) a = c d,
b = c + d, then
1 < {[F( 1,
c]/d} < 1
or, denoting the fraction by z,
1 < z < 1.
Introducing a new function
P(|z| < 1) =
f1( 1)d
(t) equal to 1 if |z| < 1 and to 0 if |z| > 1 {how about z = 1?},
f2( 2)d 2
fn ( n )
(z) d n.
Davidov does not specify that the s are independent; however [16, p. 130], according to a
contemporaneous tradition, no-one indicated this condition 3.
Noting that
I=
sinx cosxz (dx/x) =
sin[(1 + z) x](dx/x) + (1/2)
sin[(1 z) x](dx/x),
k sinu (du/u) = ( /2) if k > 0 and ( /2) if k < 0
I = ( /2) if |z| < 1 and 0 if |z| > 1
and we have
(z) = (2/ )
sinx cosxz (dx/x),
(2/ )
sinx (dx/x)
fn( n) cosxz d n.
Taking into account the Euler formula
exzi = cosxz + isinxz,
Davidov obtained
Re [(2/ )
fn( n) exzi d n],
an expression which we shall call his main formula []
z( 1,
= c + c1
+ c2
+ + cn
where the parameters are some real numbers and suppose that
)/ . Then, on the strength of (11),
i) = exp (
P(|c + c1
(1/
and fi(
+ + cn n| < 1) =
exp( y )dy + (1/
exp( y2)dy
y1,2 = (1 c)/ c1 + c2 + ... + cn .
[] If, in formula (12), c = o and the other parameters are unity, the problem is reduced to
studying the behavior of a sum of independent normal random variables.
Davidov made use of a method based on the application of the Dirichlet discontinuity
factor (above). According to Liapunov [17],
One of the most elegant among the old methods is the one based on the
application of the discontinuity factor. Glaisher [18] used it to solve the
problem under consideration.
And in 1901, in a letter to Markov, he [19] also wrote:
You are asking whether I am sure that Glaisher was the first to introduce
the discontinuity factor into the calculus of probability. I can only answer
that Laurent published his Trait du calcul des probabilits in 1873,
whereas Glaisher had published his papers in 1872. According to
Sleshinsky, Cauchy thought of applying this method, but, as far as I know,
he had not developed his idea. Leaving Cauchy out, I do not know whether
anybody had used the method of discontinuity factor before 1872.
Davidov applied that method in 1854, considerably earlier than Glaisher did.
In his article [2] Davidov explicated the principles of probability theory and its
applications for readers unfamiliar with higher mathematics. He began by considering
various random events ({the results of} drawing balls from a box or {of} tossing coins,
changes in air temperature, etc). While discussing the law of large numbers, he indicated that
the innumerable diversity of random phenomena, which, taken separately, do not obey any
definite law, reveal a surprising and unexpected regularity as the number of the phenomena
becomes ever more considerable. He illustrated this proposition by lucid and intelligible
He then posed a problem: How much can the fraction m/n derived from observations differ
from the actual probability of the studied event. He constructed an appropriate table giving
the answer sought [] by issuing from the normal distribution.
[2] A few words about the article [3]. Here, Davidov mainly repeated the contents of his
former work and extended the explication onto a more general domain, statistics, whereas
previously he had considered only medicine {medical statistics}. Here, he provided a more
precise and simple table; his tables became popular. Zhukovsky et al [15, p. 37] testified that
physicians and other readers from various places often asked him to send them a copy. In
addition, Zenetz [20] compiled a contribution based on the abovementioned papers (and
presented a copy to Davidov). With the latters permission, he reprinted Davidovs table so
as to criticize some investigations in medical statistics.
Another author included a chapter, The knowledge of the statistical method in therapy, in
his book [21]. It contained an extract from [2] with Davidovs examples replaced by
illustrations from therapeutics. Davidovs table was also reproduced there complete with
detailed explanation of its use. The celebrated physician Erisman [22, pp. 1 46] most
comprehensively explicated the contents of [2]. He set high store by it and by [5] and
reprinted Davidovs table in full. Lutzenko [23] also included an extract from the same table
and offered an example of its use. It follows that the explication of probability in medicine
was then timely and that Davidov originated work in this direction in Russia.
[3] Let us now consider Davidovs contribution to population statistics. In 1857 he
delivered a report [5] explaining intelligibly and in detail the concepts of the mean and the
probable error and the law of large numbers. He correctly maintained that means represented
nothing but a simplified and shortened expression of this law. Davidov then went on to
describe the compilation of tables of mean mortality of the male Orthodox population of
Russia for 1849 to 1853. His method consisted in comparing the number of deaths over
several years distributed by age with the number of births during those years when the died
were born.
In 1886 Davidov published a study of the laws of mortality in Russia [10]. Its first part was
a short essay where he described the history of the compilation of mortality tables and
expressed his high opinion of Buniakovsky [24]. At the same time, he justifiably criticized
the latters table since Buniakovsky had chosen 1862, the year of minimal mortality, and had
thus somewhat lessened the merits of his table. In the second part of his work Davidov
compiled a mortality table for Russia up to the age of 20 with yearly intervals. All this
testifies that he was very interested in problems of population statistics in Russia.
[4] Davidovs works in mechanics and mathematical analysis show that he adhered to
materialistic views concerning the theory of knowledge. He did not doubt that the laws of
nature were objective, he believed in the power of science closely connected with life and
technology, and he approached the study of the regularities of nature in a genuinely scientific
way. []
There exist completely opposing views on Davidovs outlook as expressed in his work on
probability. Kolman [25, p. 74] maintains that grains of materialism are to be only found in
the work of Chebyshev and his school, of Markov and others. On the other hand,
Efimochkina [14, p. 244] believes that Davidov was a progressive scientist and materialist
who clearly understood the role of probability theory and its place among other sciences.
Druzhinin [13, p. 14], basing his conclusions on only one of Davidovs contributions [8],
states that
In advancing the concept of subjective probability Davidov comes close to
Laplaces proposition according to which probability is caused partly by
ignorance, and partly by knowledge
[] Davidov [1, p. 2] thought that randomness had certain causes and that [1, p. 8] The real
difference between the so-called random and necessary phenomena consists in the difference
between their causes. At the same time, he [1, p. 6] mistakenly treated the categories of
necessity and randomness, thinking, as most contemporaneous natural scientists did, that all
phenomena and processes were necessary:
Thus, when assuming, in the accepted sense, a difference between random
and non-random phenomena, we should admit that this difference is
essentially reduced to our knowledge or ignorance of all the conditions on
which the phenomenon is dependent
Laplaces inference is undoubtedly felt here, see the beginning of his Essai philosophique sur
les probabilits.
In his course of lectures [1] as well as in his articles [2; 3] Davidov distinguished between
objective, subjective, and philosophical probabilities 4. [] The second one [1, p. 58] has
nothing in common with the phenomenon itself. Philosophical probabilities were those
based on analogies. Thus, since the Earth had some features in common with the other
planets of the Solar system, the conclusion that they were also inhabited was rather probable.
Davidov [2, p. 16] repeatedly stressed that the mathematical theory of probability
exclusively considered objective probabilities since [] only they can be precisely
determined []
[] In his later contributions [7; 8] Davidov does not mention philosophical probabilities
anymore. In 1882 he {no such date in the References} stressed that it was impossible to
apply subjective probabilities
to study the phenomenon itself, and all the mistakes made by celebrated
scientists were mostly due to their having confused subjective and
objective probabilities. A good example of such a mistake is provided by
Poissions book where he applied probability to legal proceedings.
At the time, Davidovs views concerning probability were progressive. Indeed, almost all
treatises published even at the end of the 19th, and the beginning of the 20th century defined
probability as a measure of our ignorance. []
Davidovs interest in problems in probability was incidental. He turned his main attention
to mechanics and to other branches of mathematics which is perhaps why his achievements
in probability were not as considerable. None the less, in our opinion, his methods of
investigation, especially when regarded from a historical viewpoint, are of unquestionable
interest. Appreciating probability theory at its true value, Davidov laid the basis for its
teaching at Moscow University and widely popularized it by publishing his mimeographed
courses, by delivering public lectures and putting out popular articles. []
1. For example, those of S.F. Lacroix, 1833; V.A. Jahn, 1839; Poisson, 1841, this being
the German translation of his celebrated book; G. Hagen, 1867.
2. {The first to introduce distribution functions was Poisson, see my paper in the Arch.
Hist. ex. Sci., vol. 18, 1978, pp. 252 253.}
3. {Laplace twice mentioned independence, see my paper in the same periodical, vol. 17,
1977, p. 11.}
4. {The first to introduce philosophical probabilities, unyielding to quantification, was
Cournot (Exposition de la thorie des chances, etc, 1843, Chapter 17).}
A.Yu. Davidov (Items 1 and 6 9 are mimeographed editions)
(Lectures on Mathematical Theory
of Probability). First semester. M., 1854.
2. Application of probability theory to medicine. Moskovsk. Vrachebn. Zhurnal, book 1,
1854, pp. 54 91. (R)
3. The application of the conclusions of probability theory in statistics. Zhurnal
Ministerstva Narodn. Prosveshchenia, No. 11, 1855, pp. 45 109. (R)
4. Application of probability theory in statistics. In
(Pedagogic and Literary Articles Written by Professors and Instructors of Moscow
Univ. on the Occasion of Its Centenary). M., 1855. Separate paging.
(Theory of Mean Magnitudes with Its Application to the Compilation of Mortality Tables). In
(Speeches and Report Made at the Grand Meeting at Moscow Univ.). M., 1857. Separate
paging.
% !
(Theory of Combining the Results of
Observations Leading to Least Errors, Or, the Method of Least Squares). M., 1869.
(Theory of Probability). M., 1879/1880.
(Mathematical Theory of Probability). N.p.,
10. On mortality in Russia. Izvestia Obshchestvo Liubitelei Estestvoznania, Antropologii i
Etnografii, vol. 49, pt. 1, 1886, pp. 46 66. (R)
11. Ya.I. Veinberg, Editor, Reminiscences about Davidov. Izvestia Obshchestvo Liubitelei
Estestvoznania, Antropologii i Etnografii, vol. 51, Supplement, 1887. (R)
12. Vygodsky, M.Ya. Mathematics in Moscow University in the second half of the 19th
century. IMI, vol. 1, 1948, pp. 141 183. (R)
13. Druzhinin, N.K. The notion of the law of large numbers in the Russian literature on
theoretical statistics. Nauchn. Trudy Moskovsk. Gosudarstven. Ekonomich. Inst., vol. 5, 1957,
pp. 104 122. (R)
14. Efimochkina, E.P. &
XIX .
(On the History of the Theory of Probability in Russia in the 19th Century). M., 1953. A
Candidate Dissertation. (R)
15. Zhukovsky, N.E., Nekrasov, P.A., Pokrovsky, P.M. The life and the works of Davidov.
MS, vol. 15, No. 1, 1890, pp.1 57. (R)
16. Gnedenko, B.V. On Lobachevskys work in the theory of probability. IMI, vol. 2,
1949, pp. 129 136. (R)
17. Liapunov, A.M. Sur une proposition de la thorie des probabilits. Bull. [Izvestia] AN
Psb, sr. 5, t. 13, 1900, pp. 359 386.
18. Glaisher, J.W.L. On the law of facility of errors of observation and on the method of
least squares. Mem. Roy. Astron. Soc., vol. 39, 1872, pp. 75 124.
19. Liapunov, A.M. Letter to A.A. Markov dated Oct. 1900. Archive, Russian Acad. Sci.,
Petersburg branch, Fond 173, Inventory 1, 1, No. 11. (R)
20. Zenets, M.K. The application of the theory of probability to the teaching of acute
contagious diseases. Zdorovie, No. 80, 82 and 83, 1878. (R)
(Lectures on General Therapeutics), pt 1.
21. Manassein, V.A.
Psb, 1879.
(Course in Hygiene)., vol. 2. M., 1887.
22. Erisman, F.F. &
23. Lutsenko, I.M. On the importance of probability theory in medicine. Vrach, No. 50,
1888. (R)
24. Buniakovsky, V.Ya. Essay on the laws of mortality in Russia and on the distribution of
the Orthodox population by ages. Zapiski AN Psb, vol. 8, Suppl. No. 6, 1865. Separate
paging. (R)
25. Kolman, E. On the burning importance of the theory of probability. Pod Znamenem
Marksisma, No. 2, 1934. (R)
4. B.V. Gnedenko. On Ostrogradskys Work in the Theory of Probability
Istoriko-Matematich. Issledovania, vol. 4, 1951, pp. 99 123
1. The name of our outstanding compatriot, Mikhail Vasilievich Ostrogradsky, whose
150th birthday occurred this year (he was born Sept. 12 (24) 1801 {and died 20 Dec. 1861 (1
Jan. 1862)}, is firmly included in the history of science, and, above all, in the history of
mechanics, mathematical physics and mathematical analysis. Excellent surveys of his work
concerning these directions of science had appeared [30; 20; 27; 17; 26; 28], but
Ostrogradskys work in probability theory has until now only been considered cursorily 1.
This might be explained by the fact that in this field he had only solved isolated particular
problems that remained far from the main direction of the development of the theory as
paved later on by Chebyshev. Whereas his work in mechanics, analysis and mathematical
physics are inseparably included in modern science, his results in the field of probability are
less important. Although from the viewpoint of the problems and aims of today the lack of
attention to these results is rather natural, it can hardly be completely justified since we
should also take into account whether they were interesting during his time.
The essays on Ostrogradskys mathematical work that I have seen mention only three of
his writings on probability [1; 3; 7]. One more unpublished manuscript kept at the Archive of
the Soviet Academy of Sciences is sometimes cited. There, Ostrogradsky solved the
following problem: Two gamblers decided to play a certain number of games and have
already played some games. Determine the probability that one of them wins if the
probabilities of their winning any single game are unequal.
However, two more of Ostrogradskys popular notes [4; 5] directly connected with
probability are completely forgotten. They are very interesting for achieving a more
comprehensive picture of his work and methodological views. His memoir [2] might also be
attributed to probability theory 2.
I describe the just listed six works as well as the drafts kept at the Manuscript Section at
the State Public Library of the Ukrainian Academy of Sciences; I did not study
Ostrogradskys manuscripts kept in other archives. In 1858, Ostrogradsky had delivered 20
lectures on probability theory, and there are indications that three of them were published but
I was unable to find them. Finally, A.N. Krylov 3, in an unpublished note [16] presented to
the Presidium of the Soviet Academy of Sciences with a view of bringing out Ostrogradskys
complete works, mentioned his writings on insurance. Krylov himself, as he stated, made
extensive use of them while serving on the retirement fund of the Naval Department. Here
are his own words:
In 1856, in accord with the Paris treatise, Russia was deprived of the right
to have a fleet on the Black Sea. A large number of office workers had to
be fired, and, to improve their circumstances, it was decided to establish a
retirement fund at the Naval Department, and to begin paying pensions in
1859. Life insurance was then a novelty, and calculations connected with
the work of the retirement funds, or with determining the amount of
pensions in accord with the pertinent deductions from wages were known
still less. For this reason, both mathematicians who were members of the
Petersburg Academy of Sciences, Ostrogradsky and Buniakovsky, were
included in the committee charged with drawing up a charter of the fund.
They have indeed made all the necessary calculations and provided their
theoretical justification. The transactions of the committee were published
without delay; they contain a remarkable note by Ostrogradsky and a
number of joint notes written by him and Buniakovsky.
The last sentence is inaccurate: there are no such joint papers in the transactions of the
Committee [25] which none the less contain several notes written by Buniakovsky and
Ostrogradskys note [6] (and official materials).
I shall dwell somewhat on this note. The Introduction to the transactions puts on record the
exceptionally conscientious attitude of Ostrogradsky and his colleagues towards the work of
the Committee, the importance and urgency of whose establishment I have indicated above. I
Our noted scientists, academicians Ostrogradsky, Buniakovsky, and
Veselovsky 4, were invited to participate in the work of the Committee. Not
only had they fully consented; throughout all the work, they actively took
part in the discussion and preparation of the entire project as its
permanent members.
Ostrogradskys work is best described in his own words:
We present the solution of the problem of the minimal pension and carry it
through to such simplicity that it will not cause trouble even for those least
experienced in arithmetic; it will be sufficient for them only to understand
the addition of numbers. Three tables provide the solution. []
The published tables contained the calculation of the sum accumulated after depositing 1,
2, , 9 rubles for 1, 2, , 52 years at the rate of interest of 4% (Table 1); the calculation of
the sum accumulated by payments of 1, 2, , 9 rubles yearly made in four-month
instalments after 1, 2, , 52 years (Table 2); and of minimal pensions paid out in accord
with yearly deductions of 1, 2, , 9 rubles for persons aged 41 80 years (Table 3). []
Ostrogradskys work was a valuable contribution to the theory of insurance, and retirement
funds continued to use it for a long time in their practical work.
2. {Gnedenko briefly discusses Laplaces Thorie [19] and his philosophical outlook
calling it mechanistic-materialistic.} Along with this confidence in the power of the human
mind; in its unbounded cognitive faculty; in the absence of unknowable things or
phenomena, Laplace now and then formulates idealistic propositions. In some problems he
assumes probabilities of events on the basis of the idealistic principle of insufficient reason
that consists in the following: If the probability of some event is unknown, assume as its
reasonable value a certain number according to our inner considerations 5. {Gnedenko also
quotes Laplaces celebrated statement [18, 1995, p. 2] that an omnipotent intelligence would
be able to comprehend everything.}
Obviously, we cannot say either at the first, the second, or at any other toss of a coin that
the probability of its certain face is 1/2; it is simply unknown. Its determination, the
estimation of its value, should not be done by such dubious means 6 that deprive the very
concept of probability of its objective role as a quantitative characteristic of definite real
I indicate this defect in Laplaces reasoning because Ostrogradsky followed suit. They
both perceived that the essence of the theory of probability consisted not in those logical
speculations that might be accomplished by playing upon its insufficiently definite concepts;
not in determining knowledge of a phenomenon out of its complete ignorance by means of
analytic calculations, but in being an important tool for studying regularities of a special
kind that take place in mass phenomena. They saw the justification of the theory in its ability
to solve important practical and scientific problems. Indeeed, Laplace had applied probability
for solving astronomical problems and studying population statistics, and Ostrogradsky
indicated that one of his investigations directly bore on decreasing the work necessary when
accepting delivery of large consignments at the War Department. Yes, but at the same time
they both allowed themselves such expressions as might lead to an opinion that they found
philosophical premises for using probability in insufficient reason.
Laplaces book [19] was the main source from where Ostrogradsky drew the topics for his
work in probability. Poissons (1781 1840) well-known writings [23; 24] did not
considerably influence him, because, for one thing, he obtained some results before that
French scholar did, and his scientific interests were sufficiently firmly established before
these writings have appeared.
3. Ostrogradskys work belonged to the period when physics had not yet formulated
serious problems for the theory of probability. The writings of Clausius and Maxwell were
published only during his last years, and Lomonosovs ideas about the molecular structure of
matter and his kinetic theory of heat were already all but forgotten; not only did nobody think
about their mathematical substantiation, no-one even recalled them. Applications to the
theory of artillery firing also remained in a rudimentary state. As a result, great store was
being set by applications to moral sciences 7. Laplace [18, 1995, p. 65] justified the
importance of this approach by stating that
Since most of our opinions are founded on the probability of testimony, it
is very important to submit such evidence to the calculus.
One of the most important problems facing the moral sciences was, as Laplace (Ibidem, p.
78) put it, to determine
The probability that the decision of a court which may convict only by a
given majority will be equitable, that is to say, will conform to the true
solution of the question
A large number of writings at the end of the 18th, and in the beginning of the 19th century
were devoted to solving such problems and neither did Ostrogradsky avoid them. In his first
work on probability he [1] had indeed investigated the probabilities of errors made by
tribunals. One of the main principles underpinning the solution was the assumption that the
inferences made by different judges were independent, and that they arrived at correct
decisions with one and the same probability. If a large number of judges participate in
solving an issue, then a tribunal, whose decision is reached by a majority verdict, occurs to
be, practically speaking, infallible. But the mistake in this conclusion, as I have noted
elsewhere, consists, above all, in ignoring the fact that, because of their class composition,
courts of law do not judge cases with closed eyes. In addition, as remarked by Bernstein [9,
p. 192],
This proposition is evidently wrong because it does not account for the
fact that all the judges base their judgements on the same testimonies and
material evidence. All of them will therefore gain an understanding of a
simple case more or less in the same way; however, if intricate
circumstances will mislead some of them, then a mistake will become more
probable for the other ones. In other words, independence between the
opinions of the judges is lacking, and this fact radically changes the
situation.
4. In those times, no Russian literature on probability existed at all, whereas the need to
originate it so as to attract attention to the solution of quite practical problems of the theory
of insurance and of statistics was already felt. Not without reason had Professor N.D.
Brashman, in 1841, speak about it definitely enough during a festive meeting of Moscow
University. Discussing the establishment of insurance societies in Russia, he [10, pp. 30 31]
dwelt on the need of a rigorous mathematical justification of their calculations:
Who can fail to see, with extreme regret, the absolute disregard, in the
academic institutions, of one of the most important branches of
mathematics? Only a few universities teach the rudiments of the theory of
probability, and, up to now, there does not exist a single Russian work, or
translation, on advanced or even elementary probability. True, the
publications of the Academy of Sciences include important considerations
with regard to the calculus of probability, but the rich treasures contained
there since the time of Euler are the property of the educated world
outside Russia rather than of its Russian part. We hope that Russian
scientists will soon try to make up this deficiency.
The Russian treatise [11] written by the eminent mathematician and statistician
Buniakovsky (1804 1889) only appeared in 1846. Until then, apparently only three
pertinent contributions were published in Russia: the speech of Pavlovsky[21], Professor at
Kharkov University, delivered in 1821; an article by Prince Kozlovsky [15]; and
Chebyshevs Master Dissertation [13] 8. In addition, the periodicals of the Academy of
Sciences and public journals began to publish papers by Ostrogradsky and Buniakovsky.
Chebyshevs remarkable paper [14], that might be considered as the beginning of a new
period in the development of the theory of probability, appeared in print six years after
Ostrogradskys death.
5. In 1834, Ostrogradsky [1] solves a problem already considered in a particular case by
Condorcet and Laplace: Assuming that the correctness of each judge is known, determine the
probability of passing a wrong verdict for a tribunal consisting of a given number of judges.
Contrary to Condorcet and Laplace, Ostrogradsky showed that, for judges of equal
correctness, this probability only depended on the majority of the votes rather than on the
number of judges. Later, in 1837, Poisson confirmed this result. Ostrogradsky began his
memoir by posing the following problem:
Supposing that the boundaries of the correctness of each judge are known,
the author offers analytical expressions concerning various possible cases
for the probabilities of a mistake made by a tribunal consisting of a given
number of judges. If the boundaries of the correctness are the same for
each judge [], these probabilities only depend on the majority of the
votes. [] Laplace and Condorcet thought that such a conclusion
contradicted common sense.
He then criticized Laplaces opinion
concerning the extreme difference between the probability of a mistaken
decision reached unanimously by a tribunal consisting of 12 judges and a
majority of 12 in a tribunal of 212 judges.
Ostrogradsky went on to provide his solution based on the Bayes formula and assumed
that the law of distribution of the correctness of each judge between the allowed boundaries
was uniform. He made essential use of this restriction but had not at all mentioned it. I do not
adduce his formulas since their derivation under the conditions indicated is not difficult and
the problem itself is not interesting in our time. And I have already pointed out that it was
formulated badly.
6. In his memoir [2] Ostrogradsky corrected inaccuracies made by Laplace [19]. Let yx be
a function of an integral non-negative x. Then, according to Laplace, its gnratrice
(generating function) u is determined from equality
u = yo + y1t + y1t2 + + ynt n + etc.
However, the generating function of yx+1 will not be, as he stated, u/t, but
(u yo)/t; for yx+2 it will be (u yo y1t)/ t2; and, in general, the
generating function of yx+i will be not u/ t i; as Laplace had written, but
(u yo y1t yx 1t i1)/ t i1.
It follows that the generating function of the finite difference yx will be
not [(1/t) 1]u but [(1/t) 1]u yo/t; similarly, for 2yx it will be
u[(1/t) 1] 2 (yo/t2) [(y1 2 yo)/t]
rather than u[(1/t) 1] 2, etc.
Ostrogradsky then indicated Laplaces mistake made in determining the generating
function of the sum yx and showed that the solution of finite equations became therefore
absurd. Indeed, suppose that it is required to determine yx from the equation
0 = ayx + a1yx+1 + a2yx+2 + + anyx+n
where a, a1,, an are constants. Then, as Ostrogradsky noted,
when making use of generating functions in Laplaces form we arrive at
0 = u [a + (a1/t) + (a2/t2) + + (an/t n)]
or at
0 = a + (a1/t) + (a2/t2) + + (an/t n)
from which we can determine t. However, in accord with its nature, this
variable should remain absolutely indefinite.
Ostrogradsky then pointed out that his own formula led to equality
u=
an yo + ( an 1 yo + an y1 )t + ... + ( a1 yo + a2 y1 + ... + an yn 1 )t n 1
an + an 1t + ... + at n
In concluding his note, he made known that he had applied his formulas to problems in
interpolation, extended them onto the case of two arguments, and solved many problems
reducible to finite equations pertaining to probability theory 9.
7. In 1846 Ostrogradsky, in the largest article [3] from those under our discussion, was
solving the following problem: An urn contains a given total number of white and black
balls. Some of them are drawn from the urn and their color is recorded. It is required to
determine the probability of the various possible compositions of the urn, and the probability
that the number of white balls in the urn does not exceed given bounds.
Ostrogradsky had not here betrayed his scientific principles: he considered his study as a
problem with possible important applications in everyday life rather than simply an analytical
exercise. Here are his own words:
To understand the importance of this problem, note that it will occur when
encountering difficulties connected with receiving a large number of
articles possessing certain properties, and spending some time on each of
them so as to become convinced of these properties. Military suppliers are
often obliged to carry out such work. For them, the balls in the urn are the
supplied articles; white balls, for example, are those possessing the
required properties whereas the black ones fail to have them. The drawing
of a certain number of articles {of balls} in order to record their color is
tantamount to examining a part of the supplied articles and evaluating
their quality. Fix this part, let it amount to five, six, or seven articles per
hundred, and select it randomly from all the totality. After drawing,
studying, and recording them, those that might be accepted will determine
the probability that the total number of acceptable {unacceptable} articles
will not exceed the boundaries assigned beforehand. This determination is
done in the same way as if the numbers of the white, and black balls in the
urn had to be determined. [] Thus, the solution of the problem proposed
by us might serve to diminish, approximately to 1/20, the mechanical and
most frequently very tiresome work of examining a very large number of
sacks with flour or rolls of cloth.
Ostrogradskys solution cannot satisfy us since he issued from a faulty notion that
probability was a measure of our ignorance. Consequently, he thought that probabilities were
uniquely determined even when we either would be unable to say anything about them, or
would have assumed his inference as a new hypothesis. To illustrate, we adduce the first of
Ostrogradskys arguments that he put forward when solving the following preliminary
problem: The general number of white and black balls in an urn is known, but its
composition is not; l balls are drawn. Determine the probability that, among these, m are
black and n white, m + n = l. Since, Ostrogradsky reasons, among the l balls there can be
0, 1, 2, , n, , (l 1), l
white, and, respectively,
l, (l 1), , m, , 1, 0
black balls, the probability of each of these possibilities is 1/(l + 1). He apparently had not
noticed that these possibilities were not necessarily equally probable.
He calculates a certain probability by two different methods and arrives at an equality (in
our usual notation)
m + n +1
C x C s x = C s +1
x=n
calling this result a small-scale theorem in the calculus of finite differences. The final
expression for the conditional probability that the urn containing s balls has x white balls and
y black ones, given that among the l drawn balls n are white and m black, is
C x C y C s +1 .
For calculations, Ostrogradsky recommends to use the following formula {it is extremely
involved; among other parameters, it includes the Bernoulli numbers. I am omitting it}. He
then adduces a small table of the Bernoulli numbers, considers a numerical example and
offers rather large tables for facilitating practical calculations.
8. The unfinished article [4] does not contain any analytical calculations; it is interesting
mostly in a methodological sense. Here, we once again convince ourselves that, from the
viewpoint of scientific methodology, Ostrogradsky was materialistically oriented although by
far not consistently so. We find there isolated critical remarks on the then recently appeared
book of Buniakovsky [11]. To acquaint ourselves with the nature of these objections, and to
imagine as clear as possible Ostrogradsky the popularizer, we quote his own words with
which he begins his article (pp. 29 and 30):
The theory of insurance cannot be explicated without making use of the
analysis of probability theory on which it is founded we shall therefore
try to offer a clear idea about the essence of probability, then we shall
turn to the theory of insurance and, while considering it, we shall also
speak about the principles which it borrows from the science of
probabilities.
Of course, it is possible not to speak about probabilities, and, after citing
contributions devoted to this subject, to proceed directly to the theory of
insurance. We act otherwise believing that such citations are hardly ever
pleasant for the reader, whom I would have to refer to foreign sources
since in the Russian writings known to us the theory of probability is
expounded not quite clearly or correctly.
We could have confirmed our words by a critical analysis of such writings,
but this will be out of place here. Nevertheless, begging pardon for the
digression, let us say a few words about the introduction to one of these.
The author wants to show that probability, which he calls likelihood, is a
magnitude. In proving this, he attempts to convince us that some
likelihoods can be higher or lower than other ones. Then, after offering all
the arguments, sufficient, according to his opinion, for complete
conviction, he infers that likelihood, just like any other mathematical
magnitude, should be measured and admits of a measure.
Thus, the learned author believes that likelihoods are mathematical
magnitudes only because some of them can exceed or be less than other
ones. This opinion is not quite correct. Indeed, do not we really say, and,
while saying it, do not we clearly understand, that such-and-such a
scientist is more conscientious than another one; that a Frenchman is
braver than a German; that a reader is more sensible than an author, etc.
Conscience, bravery, wisdom can therefore be larger or smaller and are
mathematical magnitudes, they can be measured, expressed in numbers,
and various operations might be carried out on them. By such reasoning
we could have considerably widened the scope of the mathematical
sciences; fundamentals of mathematical theories of unscrupulousness,
absurdity, etc can thus appear.
This sharp criticism was undoubtedly directed against Buniakovsky [11] 10, although it
ought to be said, not against its main defect, which was the authors subjective approach to
probability: Ostrogradsky himself, as we shall see now, was guilty of the same. He
The word probability has no meaning in itself, it always concerns some
phenomenon or event. A phenomenon is usually called probable when
believing that it will rather happen than fail to occur. On the contrary, a
phenomenon is considered improbable when the reasons for its failure are
thought to be more convincing than those favoring it.
We note in passing that the analysis and estimation of the circumstances
accompanying the phenomenon can be very different for different persons.
A phenomenon considered probable, or even certain by one man will be
thought improbable by another one. Almost all wagers, games of chance,
etc take place because of such differences in opinion. We say almost
because bets are sometimes meant to be lost to a needed high-ranking
Geometers attach another meaning, different from the usual one [] to
the word probability. For them, anything whose impossibility or certainty
is not rigorously proved is probable. Thus, in the eye of geometers even a
phenomenon, that all other people will consider unusual, might possess
some probability; and, on the contrary, for a geometer, a phenomenon
considered certain might be only probable. True, it will, generally
speaking, be highly probable.
Nature has no probabilities 11. All that occurs in nature is doubtless and
certain. Probability is a corollary of mans weakness; it concerns us,
exists for us and can serve only us. Its study is an important, and even a
necessary supplement to those few truths that we know with a relative
certainty.
If a phenomenon depends on several other phenomena or cases, some of
them producing it, and the others opposing it, and if, furthermore, they are
such that for us, we repeat: for us, there is no reason for preferring
some of them to the other ones, then the probability of the expected
phenomenon is measured by a fraction []. It follows [] that we can
only determine the probability of such phenomena that can be directly or
obliquely separated into cases in whose occurrences we do not perceive
any preferences of one over another.
Five balls are in a bowl. [] We do not know why one ball will be drawn
rather than another one. A reason, about which we speak, undoubtedly
exists, but it is absolutely unknown to us, and, since we are unable to
prefer one ball over the other ones, all of them present for us equally
possible cases. He, who would have known the arrangement of the balls in
the bowl, and would have been able to calculate the movement of the hand
drawing a ball, he would have said beforehand which ball will be
extracted; for him, there would have been no probability.
For Ostrogradsky, therefore, probability is nothing but the cognizing subjects measure of
certainty. All the inferences of probability theory are thus deprived of the objective essence
independent of that subject. It is interesting to note that at the same time Ostrogradsky, when
dealing with concrete practical problems, completely renounces his subjectively idealistic
interpretation of probability, and regards it as a quite objective numerical measure of
phenomena of a certain kind. It is sufficient to recall, for example, his belief in a real
possibility of reducing to 1/20 the tiresome work of examining the quality of delivered
articles to convince ourselves that, concerning problems in natural sciences, Ostrogradsky
was a materialist a spontaneous materialist, as we would say now.
In the first part of his paper Ostrogradsky argues with the followers of Saint-Simon who
had declared, as is seen from a passage quoted by him, that if all the balls were perfectly
similar and the bowl was such that all of them had equal chances of being drawn, then no ball
will ever be drawn, and the entire theory of probability becomes therefore senseless. Here we
encounter, in another setting, the generally known story about the Buridans ass. We shall not
repeat Ostrogradskys critical remarks since they occur, in essence, in the previous passages.
In the second part of his article Ostrogradsky offers a popular exposition of the concept of
expectation. To illustrate, he asks:
Suppose that you obtained a lottery ticket, of course not for free; it costs
something, and at the same time you bought a hope of winning. But did not
you pay too much? It wouldnt be bad to know how much does it {really}
cost.
And he adds ironically:
We note in passing that the newspaper advertisements announcing lotteries
always omit an important circumstance without which you have no means
to estimate your hopes of winning. You usually read: The lottery has so
much articles of excellent quality to be won, the tickets costing so much
can be obtained in such-and-such a place. The total number of tickets is
never mentioned, and without it you are unable to calculate the probability
of winning, i.e., you cannot determine an element required for estimating
the chances. Thus, you pay for a hope of winning not knowing its price,
and you are always paying in excess.
After offering such introductory explanations of the meaning of expectation illustrated by
lotteries and the expected winnings, Ostrogradsky goes over to insurance. He had not
achieved much: he posed a question about the methods of calculating the cost of insuring an
article, considered a numerical example and, having determined this cost in one case, advised
his readers:
Do not pay more; on the contrary, try to pay less so as to gain something.
Do not worry about the insurance society: it will not incur losses.
I was unable to find the promised sequel.
9. The note [5] is similar in essence to the paper on insurance [4]. It is also devoted to
explaining to the general public that all kinds of chance undertakings, lotteries, games, are
disadvantageous to them, but fetch a sure and easy profit for the businessman in charge.
Somewhat earlier, in 1840, Buniakovsky [12] expressed the same idea: a lottery, he argued,
was a tax levied on ignorance 12.
Ostrogradsky considered in detail the game of craps, where several dice are thrown on a
table and the winner is determined by the number of points achieved. He calculated the
probabilities of getting various scores in a throw of two, three, , twelve dice, and on an
adduced plate he offered an appropriate table. Ostrogradsky then asked his readers: In
Pavlovsk, you have undoubtedly seen this game played with eight dice. Would you like to
find out whether it is profitable? He did not know the going payouts and restricted his
explanation with purely theoretical calculations and an introduction of a simplest idea of
expectation. He concluded by formulating a rule for determining the most profitable (the
most probable) number of points. [] Ostrogradskys inference is the same as the one he
vividly expressed earlier [4]:
You say that such-and-such had won the main prize and all at once gained
a fortune so why wont I also win the same prize? You are unwittingly
comparing yourself with the winner, and do not want to admit that it is
much more natural to place yourself among the losers since they are by
far more numerous.
10. In 1858, the Scientific Council of the Mikhailovsky Artillery School, where
Ostrogradsky taught for many years, resolved to establish optional courses to help cadets
develop scientific initiative and widen their mental outlook. Ostrogradsky announced an
elective course in the theory of probability. A brief report about it is to be found in [22, p.
265], and we quote:
Hundreds of cadets gathered for the first lecture of the celebrated
professor. Ostrogradsky, well dressed in a tail-coat and wearing his order,
described to his audience, with singular grace and remarkable simplicity,
making hardly any use of the blackboard, the origin of probability theory
and its principles. Everyone was delighted and all at once several listeners
expressed their wish to write down and print the lectures. The second
lecture was still attended by very many cadets, but this time Ostrogradsky
had to use chalk and he was less enthused. The third one gathered a small
number of listeners, only five attended the fourth one, and this number did
not increase. Three persons were present on the last (the twentieth) lecture
and the printing of the notes was discontinued after the third one.
That the number of listeners had sharply declined towards the end of the course is a rather
usual phenomenon that also takes place in our time. For us, it is essential that three of the
lectures were printed, apparently by a lithographic process. I have not seen this edition that
deserves a detailed study because the first lectures are especially interesting in the
methodological sense, and, besides, because they allow to judge to a certain extent the scope
of the entire course. It is likely that the brief historical essay and some general reasoning on
probability kept at the State Ukrainian Public Library [8, sheet 904] indeed constitute an
outline of the introductory lecture of 1858. Since it is of possible interest, I reproduce its
subject-matter in full. I invented some words which I was unable to decipher in accord with
the context and placed them in brackets.
The theory of probability ought to be attributed to the sciences of the
modern times since its real beginning does not go back further than the
mid-17th century. True, some subjects belonging to this science have been
known in very remote times: calculations based on mean longevity were
made continually; marine insurance was practised, the number of chances
in games of chance, although only in the simplest of them, were known;
and the stakes in fair games were determined.
However, such conclusions were not subordinated to any rules. []
Before the mid-17th century problems concerning probabilities were not
subjected to mathematical analysis, and no rigorous general rules for
their solution had existed.
Pascal, and, following him, Fermat, geometers of the 17th century, are
justly considered the founders of the science of probability. The first
problem belonging to this science was rather difficult. {Ostrogradsky
describes the problem of points.} It is remarkable that the name of
Chevalier de Mr, a man of the world with no [success in the field of
B.G.] mathematical sciences, remains forever in their history.
However, we ought to deal not with the history of the calculus of
probability, but with the science itself. The first question presents itself
here: What should be meant by probability? Let us analyse several phrases
containing this word and see what sense should be assigned to it.
Tomorrow, it is said, the weather will probably be fine. []
Here we again encounter the same methodological approach to the concept of probability
as we already saw in [4].
11. Ostrogradskys last article on probability theory [7] was published in 1859, i.e., at the
very end of his creative life. It illustrates how a more general point of view simplifies
exposition, and how a problem, whose elucidation in former times had taxed the wits and
efforts of the best minds, is clarified to the utmost. Nowadays the Bayes formula belongs to
the most elementary results of the theory; it is derived almost trivially when issuing from the
definition of conditional probability and the formula of total probability 13. In Ostrogradskys
times, however, it was necessary to consider numerous cases, and to formulate principles,
only then passing onto the derivation of the formula itself in one or another form, under some
conditions or other. Ostrogradskys article was indeed devoted to deriving this formula.
He indicated that Laplace had considered without proof only the case in which the prior
probabilities of the hypotheses coincided and put forth this formula as a principle. Later
Gauss proved Laplaces principle under the same conditions {no reference provided}14.
Ostrogradsky derived the Bayes formula without assuming that the prior probabilities of the
hypotheses coincided. In concluding, he remarked that Poisson had also proved the same
formula but that that proof seemed to him insufficiently straightforward.
12. Among Ostrogradskys manuscripts at the State Ukrainian Public Library [8] there are
pages devoted to probability theory, and we have fully reproduced the essence of one sheet.
Sheets 443 466 contain drafts of [3]. In addition to calculations included in its final text,
there are many pages with numerous uncompleted attempts at proving the identity of some
expressions. Other pages contain initial formulations, problems, a study of the change of the
probability depending o some parameters (this is also included in the final text), an outline of
a derivation of the main formula of 7 {omitted there} as well as numerical examples. Sheets
520 525, united by a general heading Probabilities of future events, only contain fragments
which apparently preceded the compilation of [7].
13. We have examined Ostrogradsky works in the field of probability and became
convinced in that all of them were devoted to subjects then strongly exiting science. A
correction of Laplaces mistaken conclusions and a derivation of results later obtained by
Poisson testify to a sufficient extent that also in probability, although this discipline was
situated on the borders of his scientific interests, Ostrogradsky kept abreast with his time.
[] His work on probability provide an additional feature for scientifically characterizing
him and reveal the breadth of his interests []
1. {General sources concerning Ostrogradsky also include Gnedenko (1951; 1984) and
Maistrov (1974, pp. 181 187) who closely follows the paper now under translation and
reprints a number of quotations from Ostrogradskys writings. Seneta (2001) contributed a
new paper on my subject. References to my Notes follow after their main list below.}
2. {Ostrogradsky also attempted to generalize the concept of moral expectation; the only
relevant source is Fuss (1836, pp. 24 25), see my paper in the Arch. Hist. Ex. Sci., vol. 16,
1976, pp. 170 171.}
3. {Aleksei Nikolaevich Krylov (1863 1945), a naval architect and applied
mathematician.}
4. {Apparently Konstantin Stepanovich Veselovsky (1819 1901), an economist and
statistician.}
5. Laplace (Trait de Mc. Cl., t. 3, p. xi; my paper in the Arch. Hist. Ex. Sci., vol. 17,
1977, p. 5) thought that hypotheses should be rectifiant sans cesse par de nouvelles
observations, etc. It is hardly reasonable to criticize classics without a thorough preliminary
research.}
6. {Gnedenko did not elaborate.}
7. {Many authors had been opposing the application of probability to jurisprudence; Mill,
in 1843, called it an opprobrium of mathematics and Poincar stated that, in law courts, men
behave like the moutons de Panurge; see Gnedenko & Sheynin (1992, p. 280n) and my paper
in the Arch. Hist. Ex. Sci., vol. 9, 1973, p. 296, with a description of pertinent events in the
mid-19th century. Note also that Laplace [19, p. 523] (and obviously Poisson) considered the
case of independent jurors; they apparently thought of investigating the ideal conditions, of
estimating the utmost possibilities of law courts. Then, contrary to Gnedenkos context (also
see end of his 5), such applications of probability are nowadays being resumed (see Heyde
& Seneta 1977, p. 34), who provide several references, and Eggleston 1983).}
8. {Also Zernov (1843).}
9.{I specify a few points. First, the memoir was lost and only its Extrait was published.
Second, Ostrogradsky provided an exact reference to Laplace [19, pp. 9ff]. Third, he indeed
showed that Laplaces mistake could have led to absurdity, but his example was invented
rather than borrowed from [19]. In 1959 1961 Ostrogradskys '
(Complete Works) were published in Kiev in three volumes and the Exrait was included in
vol. 3 with a commentary by Gnedenko (pp. 347 348). Yes, Gnedenko mentioned that the
memoir itself was lost, but he did not specify the other points, see above.}
10. {In effect, Buniakovsky went back on his own words and adopted an objective
approach; see my paper in the Arch. Hist. Ex. Sci., vol. 43, 1991, p. 202.}
11. {This statement is much too restrictive; nature has both necessity and randomness.}
12. Cf. Petty (1662, p. 64): A lottery [] is properly a Tax upon unfortunate self-conceited
13. {Gnedenko had not dwelt on the importance of the Bayes formula and approach or on
the pertinent protracted debates in the 20th century.}
14. {In 1809, Gauss justified the principle of inverse probability for the case of equal
probabilities of the various hypotheses.}
M.V. Ostrogradsky
I mentioned his Complete Works in Note 9, see above.
1. Extrait dun mmoire sur la probabilit des erreurs des tribunaux. Bull. Scient., No. 3,
1834, pp. xix xxv. The Bulletin was appended to the Mm. Acad. Imp. Sci. St.-Ptersb.,
6me sr., Sci. math., phys. et naturelles, t. 3, this being Sci. Math. et phys, t. 1.
2. Sur le calcul des fonctions gnratrices. Bull. Scient.AN Psb., t. 1, No. 10, 1836, pp. 73
3. Sur une question des probabilits. Bull. Phys.-Math. AN Psb., t. 6, No. 21 22, 1846,
pp. 321 346.
4. On insurance. Finsky Vestnik, vol. 13, No. 1 1847, pp. 29 34; No. 2, pp. 40 44. (R)
5. Dicing. Ibidem, No. 3, pp. 29 32. (R)
6. Note on the retirement fund [25, pp. 93 101 + Tables]
7. Sur la probabilit des hypothses dprs les vnements. Bull. Phys.-Math. AN Psb, t.
17, No. 33, 1859, pp. 516 522.
8. Manuscripts kept at the manuscript section, Ukrainian State Public Library,
Ostrogradskys Papers.
9. Bernstein, S.N. (1946),
(Theory of Probability). Fourth ed. M.
10. Brashman, N.D. (1841), (
(On the Influence of Mathematcal Sciences on the Development of
Intellectual Faculties). M.
11. Buniakovsky, V.Ya. (1846), (
(The Principles of the Mathematical Theory of Probability). Psb.
12. --- (1840), Thoughts about some unfounded notions in social life. Maiak, pt. 3, pp. 80
94. (R)
13. Chebyshev, P.L. (1845), (
(Essay on the Elementary Analysis of the Theory of Probability). '
(Complete Works), vol. 5. M. L., 1951, pp. 26 87.
14. --- (1867), Les valeurs moyennes. J. math. pures et appl., t. 12, pp. 177 184.
15. Kozlovsky, Prince (1836), On hope. Sovremennik, vol. 3, pp. 23 47. (R)
16. Krylov, A.N. (Manuscript), To the memory of Ostrogradsky. Rossiysk. Akad. Nauk,
Archive, V, Inventory 1 0, No. 11. (R)
17. Lakhtin, L.K. (1901), Ostrogradskys work in analysis.MS, vol. 22, No. 4, pp. 540
554. (R)
18. Laplace, P.S. (1814), Essai philosophique sur les probabilits. Oeuvr. Compl., t. 7, No.
1. Paris, 1886. Separate paging. English translation : Philosophical Essay on Probabilities.
New York, 1995.
19. --- (1812), Thorie analytique des probabilits. Oeuvr. Compl., t. 7. Paris, 1886.
20. Liapunov, A.M. (1902), Ostrogradskys merits in mechanics [29, pp. 115 118].
(On Probability). Kharkov.
21. Pavlovsky, A.F. (1821), (
22. Platov, A., Kirpichev, L. (1870),
$ (Historical Essay on the Establishment and
Development of the Mikhailovsky Artillery School). Psb.
23. Poisson, S.D. (1837), Sur la probabilit du tir la cible. Mmorial de lartill., No. 4,
pp. 59 94.
24. --- (1837), Recherches sur la probabilit des jugements, etc. Paris.
25. Anonymous (1858), '
(Proposal
on Establishing a Retirement Fund at the Naval Department). Psb.
26. Sabinin, E.F. (1901), Ostrogradsky. MS, vol. 22, No. 4, pp. 499 531. (R)
27. Steklov, V.A. (1902), On the work of Ostrogradsky in mathetical analysis [29, pp. 118
127].
28. Tikhomandritsky, M.A. (1902), An essay on Ostrogradskys work in pure mathematics
[29, pp. 92 115].
29. Tripolsky, P.I. (1902), M.V. Ostrogradsky. Poltava. (R)
30. Zhukovsky, N.E. (1901), On the scientific work of Ostrogradsky in mechanics. '
(Complete Works), vol. 9, 1937; vol. 7, 1950. M. L.
References to Notes
Eggleston, R. (1983), Evidence, Proof and Probability, 2nd edition. London.
Fuss, P.N. (1836), Compte rendu de lAcadmie pour lanne 1835. In Recueil des actes
de la sance de lAcad. Imp. Sci. St.-Ptersb. 29 Dec. 1835. Psb. Separate paging.
Gnedenko, B.V. (1951), Ostrogradsky. Uspekhi Matematich. Nauk, vol. 6, pp. 1 25. (R)
--- (1984), Ostrogradsky. M. (R)
Gnedenko, B.V., Sheynin, O. (1992), Theory of probability. In Mathematics of the 19th
century, vol. 1. Editors, A.N. Kolmogorov, A.P. Youshkevich. Basel, pp. 211 288.
Originally published in Russian in 1978.
Heyde, C.C., Seneta, E. (1977), I.J. Bienaym. New York.
Maistrov, L.E. (1974), Probability Theory. Historical Sketch. New York. Originally
published in Russian in 1967.
Petty, W. (1662), Treatise of taxes and contributions. In Economic Writings, vol. 1.
Cambridge, 1899, pp. 1 -97.
Seneta, E. (2001), Ostrogradsky as probabilist. Ukrainsk. Matematich. Zhurnal, vol. 53,
pp. 1038 1046.
Zernov, N. [E.] (1843),
(Theory of probability). M.
5. S.N. Bernstein. Chebyshevs Work in the Theory of Probability (1945).
It is obvious that Bernstein had not studied the early history of probability. This apparently
goes to show the general situation, at least in Russia in 1945, when he published his paper,
and perhaps in 1964, when it was reprinted. Several of my Notes indicate his pertinent
particular mistakes and shortcomings.
1. The number of Chebyshevs writing in probability is not large. There are only four of
them: his Master Dissertation [1] followed by a paper [2]; then, after more than 20 years, an
article [3] published in two periodicals at the same time; and, finally, after another 20 years,
his last writing [4]. Papers [2] and [3] are reprinted in vol. 1 (1899) of his collected works
published by the Petersburg Academy of Sciences, and article [4] is in vol. 2 (1907) of the
same source. Neither his dissertation [1], nor his course of lectures in probability read at
Petersburg University from 1860 to 1882 and published only recently from the extant notes
taken in 1879 1880 by his illustrious student, Liapunov, were included in this collection.
That Chebyshev, barely graduating from the [Moscow] university, chose probability as the
subject of his Master Dissertation, testifies that he became especially interested in this
peculiar branch of mathematics from his earliest years.
2. At its origin, the theory of probability is known to have been far from the general
movement of the natural sciences, and the only experiments by whose means its most
important notions and main principles had been developed and specified, were provided by
games of chance. It was on these grounds that Jakob Bernoulli, more than 200 years ago,
discovered his celebrated theorem that offered a clue to understanding the origin of mass
regularities out of independent individual fortuities and represented the first rigorously
proved, although very particular, formulation of the law of large numbers.
Laplaces classical researches, and, above all, the De Moivre Laplace limit theorem,
constituted the next stage in the development of probability theory. This proposition
established the limiting law of probability
( xa) /
exp ( z2)dz
G [(x a)/ ] = (1/ 2 )
for the deviations of the number (x) of the occurrences of some random event E from the
expectation a = np of x in n Bernoulli trials with E having a constant probability p [and the
variance 2 of x being {now} known to equal np(1 p)]. The law of large numbers and the
Laplacean normal law G(x), both of them being of fundamental methodological importance,
were thus established in their simplest forms already in the 18th century 1. Another of
Laplaces considerable merit was that he had foreseen the extreme generality of these laws
and led the theory of probability out from the narrow confines of games of chance onto the
vast domain of scientific natural sciences 2.
Owing to Laplaces influence, the first half of the last {of the 19th} century was marked by
a heightened interest in probability and emotion for its applications. Many of these were,
however, insufficiently justified; and some, even those supported by Laplace himself and
Poisson, were so obviously mistaken that Mill later on well-deservedly qualified them as a
mathematical opprobrium 3.As a result of ensuing failures {?}, the enthusiasm gave place to
disappointment and West-European mathematicians were becoming ever more convinced
that the theory was only a mathematical entertainment of sorts not admitting of essential
scientifically substantiated applications and hardly warranting the attention of serious
scientists.
3. Chebyshev regarded the crisis point reached by probability theory in another way.
Owing to the practicality natural to him, and without dwelling on justifying it as a method of
scientific research by deep philosophical considerations (that only became possible in our
time), he understood that, since there existed real random phenomena (games of chance) that
confirmed with maximal precision its simplest mathematical inferences, the substance and
the more general conclusions of the theory should find practical applications. To illustrate:
the applicability of geometric formulas to the calculation of the area of a rectangle leads to
their applicability to the cases of more complicated but precisely enough determined flat
It was only necessary to construct, within the context of the same main notions, more
flexible but precisely defined theoretical patterns and to adjust them to the properties of the
random events and magnitudes observed in reality. As he said in his dissertation,
The science of probability, that goes by the name of the theory of probability, has as its
subject the determination of the probability of an event given its {the events} connection
with events whose probabilities are known.
In particular, for Chebyshev the pivotal task was the solution of the two principal problems
on which the subsequent destiny of the theory depended: To formulate precisely, and to
prove rigorously for the widest possible class of random phenomena, first, the law of large
numbers; and, second, the limit theorem for sums of independent random variables.
Chebyshev formulated these problems, by no means equally difficult, while in his first youth.
He brilliantly solved the first of them [3] whereas the second one remained the object of his
deep thoughts to the end of his life, and, in essence, he completely solved it only in 1887 [4].
We shall now try to outline, as far as it is possible when studying his published works, the
main stages of Chebyshevs development as a specialist in probability.
4. We shall not dwell on Chebyshevs first paper [1]; it is important only in its goal which
was to formulate precisely the general theorems of the theory of probability and to prove
them while advancing to the forefront inequalities and the estimation of the error of limiting
formulas. For Chebyshev, the law of large numbers as well as the limiting normal law thus
made sense not as mathematical characteristics of some infinite sets, but as approximations to
quantitative relations observed in sufficiently numerous and really existing structures of
some random elements. And it was essential that the word sufficiently be quite definitely
interpreted in a mathematical way.
The main principles of the theory of probability as indicated by Chebyshev in the
beginning of his dissertation, which did not differ in essence from those adopted by his
predecessors, are reduced to the following:
If, under known circumstances, out of a definite number of different events one must
necessarily occur, and there is no special reason for expecting some of these events rather
than the other ones, then we distinguish such events by calling them equally possible
occasions. Thus, 1 and 0 are the limits of the probability of events; it reaches the former by
increasing in cases of necessary events; and it reaches the latter by decreasing in cases of
impossible events. For all other events, having neither necessity nor impossibility,
probability remains different from 1 and 0. We consider it approximately doubtless that
events will, or will not take place if their probabilities differ but little from 1 or 0. Such are
all the conclusions inferred from observations and testimonies.
Chebyshev undoubtedly saw well enough the inadequacy of the elementary algebraic
method that he applied. In particular, he [1] proved the Poisson theorem (the law of large
numbers) only for a restricted number of different probabilities. While seeking new
approaches, he apparently only discovered a general elementary proof of this theorem
complete with an appropriate estimation of error when [1] was being printed. The
substantiation represented the first example of an extremal reasoning characteristic for his
entire later work. It was contained in Chebyshevs article [2] 4 that is a valuable original
supplement to his dissertation. I adduce the typical for Chebyshev beginning of this article:
The subject of this note will be the proof of the following proposition: It is always possible to
assign such a large number of trials 5 that the probability that the ratio of the number of the
occurrences of some event E to the number of trials will not deviate from the arithmetic mean
of the probabilities of E beyond given boundaries no matter how close they are one to
another, and that that probability will be arbitrarily close to certainty.
This main proposition of probability theory that includes as a particular case the Jakob
Bernoulli law, was derived by Poisson [19, Chapter 5] who issued from approximate
calculations of some rather complicated definite integral. However, regardless of the
cleverness of the method used by the celebrated Geometer, it does not provide the limit of the
error allowed by his approximate analysis, and because of this uncertainty in the value of the
error the demonstration does not possess appropriate rigor.
Chebyshevs proof is based on the following remark. The probability Pm that the event E
will occur not less than m times in trials is equal to some expression, symmetric with
respect to all the pk, k = 1, 2, , which are the probabilities of E at trials k, and linear in each
pk, for example, with respect to p1 and p2. Thus,
Pm = U + V(p1 + p2) + Wp1p2
where U, V, W do not depend on p1 or p2 anymore. Therefore, if 0 < a =
p1 + p2 1 then
Pm = U + Va + Wp1p2 takes its maximal value either at
p1 = p2 = a/2, or at p1 = 0. The case of a > 1 is reduced to the previous one by replacing the
expression of Pm by a similar formula including q1 = 1 p1 and
q2 = 1 p2. A theorem of a specifically Chebyshev type is now easily derived: The maximal
value of Pm corresponds to such values p1, p2, , p that
p1 = p2 = = p = 0, p +1 = = p + = 1,
pi = (S )/[( ( + )], + < i , p1 + p2 + + p = S,
0 are some definite numbers. Thus,
( )!
( m )!( m )!
+ ... .
m +1 S
The terms of the sum decrease more rapidly than a geometric progression with ratio
replacing them by such a progression, we find that
Pm <
m +1
Chebyshev then notes that, if m > S + 1, the right side of this inequality increases with the
decrease of the integers > 0 and > 0, so that, assuming that = = 0, we derive the
simple inequality
m m/(m
S)
where m is the probability that the number of the occurrences of the event E is m when pi= p
= S/ (i.e., when the Poisson pattern is replaced by appropriate Bernoulli trials). After a
similar estimation of the probability that E will occur not more than m < S 1 times, the
demonstration is easily concluded. Thus, the very precise value of the lower boundary,
derived in [1] for the number of trials sufficient for the probability of the deviations
exceeding the given limits to be lower than some arbitrarily small magnitude, can be applied
to the Poisson general case.
The idea about the existence of an inequality of type (1) might have possibly occurred to
Chebyshev in connection with the fact that could have hardly escaped his notice, that for
independent trials corresponding to a given mean probability p = S/ the variance of m is
maximal if all the probabilities are equal one to another. Be that as it may, the peculiar clever
demonstration of this inequality was only an elegant episode in Chebyshevs work,
unconnected with the method of expectations or moments from which he was then far and
which he created much later when solving concrete problems belonging to other branches of
mathematics. Laplaces transcendental methods did not satisfy him whereas the usual
algebraic methods occurred to be too weak for enabling to extend essentially the domain of
reliable applications of the theory of probability.
5. During the next years after defending his Master Dissertation Chebyshev is known to
have written a number of remarkable works on integrating algebraic functions by means of
elementary functions and to have immortalized himself by solving classical problems on the
distribution of primes that did not yield to the efforts of greatest mathematicians.
The solution of problems concerning algebraic integrals led Chebyshev to a deep study of
the properties of algebraic continued fractions which soon became his favorite tool,
exceptionally powerful and fruitful in his hands. However, it is difficult to assume that
already then Chebyshev had foreseen how he will use it in probability. Such an idea might
have first crossed his mind after he, in 1855, had applied continued fractions to interpolation
by the method of least squares [6]. It would have been an essential stretch of imagination to
attribute Chebyshevs works on the method to probability. Their direct aim was to change
expediently the technique of calculation so as to specify when required the obtained
approximate expressions in the simplest and shortest way; the formal algebraic
transformations do not possess here any stochastic meaning. For Chebyshev, the deep
theoretical essence of interpolation by least squares consisted not in its possible connection
under more or less arbitrary assumptions with probability; for him, it was most important that
the method provided a natural constructive approach to expanding an arbitrary empirical
function in a series of polynomials converging (in some sense, in a best way) over a given
interval of any length whereas the Taylor Maclaurin series only sufficiently approximates
functions at small values of the independent variable.
Owing to his instrument of continued fractions, Chebyshev went by algebraic means far
beyond classical algebra and entered the immense field of the general theory of functions.
The entire domain of the unreliable transcendental methods of analysis now became
accessible to his precise trustworthy methods of augmented algebra. And, by their means, he
must {now} appropriately formulate and solve in all algebraic rigor the abovementioned
main problems of the theory of probability.
Indeed, a parabolic interpolation of a function (x) given its values at m points xi by least
squares is known to consist in determining a polynomial Pn(x) of degree n < m + 1 under the
condition that the mean square error (with assigned weights 2(x))
In2 =
[Pn(xi) (xi)]2 2(xi),
i =1
(xi) = 1
be as small as possible. For a finite m the minimal value of In2, decreasing with the increase
in n, vanishes when n = m 1 and Pn(x) then becomes the Lagrange interpolation
polynomial.
Chebyshevs essential and practically very useful simplification of the solution of this
algebraic problem, on whose details we shall not touch, consisted in that he represented the
Lagrange interpolation polynomial Pm1(x) as
Pm1(x) =
Ak k(x)
k =0
where k(x) were such polynomials of degree k m 1 that for any n < m 1 the
Pn(x) =
ensured the minimal value of In2. It occurred that this property was possessed by the
denominators i(x) of the consecutive convergents of the continued fractions emerging from
(xi)/(x xi)
which were thus orthogonal polynomials with weights
s(xi)
i(xi)
= 0, s * i.
(x). In other words, they obeyed the
Chebyshev soon extended his conclusions onto the case of m
so that the finite sum
(2) was replaced by an integral. Solving the problem of bringing the integral
In =
[Pn(x) (x)]2 2(x) dx
to its minimal value in a similar way and basing his work on the algorithm of continued
fractions, he created the general theory of expanding an arbitrary function in a series of
orthogonal polynomials. Thus, it is impossible not to admit that Chebyshev was the founder
of this central direction of the modern theory of functions. True, his expansions in the most
important orthogonal polynomials were largely formal and he did not appropriately examine
the conditions for their convergence. By analogy with a finite m, when the minimal value of
the mean square error In2 vanishes at n = m 1, Chebyshev argued without proof that in his
also in the case of (6).
examples the same happened as n
6. Chebyshevs general ideas described above came into contact with probability theory in
his paper [7] read in 1859. There, for the first time, he provided the expansion of an arbitrary
function F(x) on the entire real axis into polynomials
l(x)
= exp (kx2) [d l exp ( kx2)/dxl]
which, as he showed, were the denominators of the convergent continued fractions
exp( ku 2 )
F(x) =
Al l(x),
l =0
l!(2k) Al =
exp( kx2) l(x)F(x) dx.
If f(x) = F(x) exp ( x2/2), the series (9) becomes the density of any normed law
xf(x) dx = 0,
x2f(x) dx = 1
f(x) dx =
f(x) = [exp ( x2/2)/ 2 ][1 +
Al l(x)].
After integrating term by term we obtain
f(x) dx = (1/ 2 )
exp ( x2/2) [1 +
Al l(x)] dx.
(10bis)
Chebyshev applies this formula in his memoir [4] where the coefficients Al are expressed
in terms of the moments of
Xn= (1/ Bn )
ui,
of the normed sum of independent variables ui. It is therefore natural to assume that, after
discovering the formal expansion (10bis), Chebyshev already in 1859 more or less clearly
saw the approach to proving the limiting theorem of the theory of probability. The
demonstration should have established some boundary for
(1/ 2 )|
exp ( x2/2)
Al l(x) dx|
tending to zero when the number n of the terms ui infinitely increases. Since the coefficients
Al = (1/l!)
l(x)f(x)
are completely determined by the moments (i.e., by the expectations of Xnk, k l), it is also
obvious that the latter acquire special importance. However, Chebyshev was certainly fully
aware that the analysis that confronted him was of the same order of difficulty as the problem
of imparting necessary rigor to Laplaces transcendental methods.
7. From 1860 Chebyshev took over from Buniakovsky the teaching of probability theory at
Petersburg University which apparently imparted an additional stimulus for directing his
thoughts to this field. It seems, however, that for a number of years the limit theorem did not
yield to his efforts and remained an unattainable goal. Much later, in his course [5],
Chebyshev explicated the Laplace method without mentioning the method of moments and
turned the students attention to its inadequacy:
In its present state, mathematical analysis cannot derive this boundary {of the ensuing error}
in any satisfactory fashion.
On the contrary, in 1866 Chebyshev brilliantly solved the first problem which was to
provide a general elementary proof of the law of large numbers easily understood by average
students. Introducing absolute clarity into mathematical definitions and reasoning into
probability, and, in particular, precisely establishing the general properties of expectations,
he was able to arrive at his celebrated derivation of the so-called Chebyshev inequality from
which directly followed the classical formulation of the law of large numbers:
If the expectations of the squares of [independent]variables u1, u2, u3, do not exceed a
given finite boundary, the probability that their arithmetic mean N differs from the arithmetic
mean of their expectations less than by some given magnitude tends to unity as N increases to
Chebyshev reported this finding to the Academy of Sciences in 1866 and published it next
year [3] in two periodicals. One of these carried, in the same volume, a reprint of a memoir
by the French mathematician Bienaym that included an inequality similar to the Chebyshev
inequality written out in a somewhat less general form. This fact to some extent deprives our
great compatriot of priority. Markov [17d, 1924, p. 89] used the term Bienaym Chebyshev
inequality and justified it in the following way (p. 92):
We connect the two names, Bienaym and Chebyshev, with this remarkable simple inequality
since Chebyshev first clearly formulated and proved it, but Bienaym had much earlier
indicated the main idea of the demonstration. The inequality itself, only accompanied by
some particular assumptions, can be found in his memoir [12].
Bienayms memoir aimed at defending Laplaces stochastic justification of the MLSq
which Cauchy had called in question. In getting to know Bienayms work, Chebyshev
apparently found ideas akin to his own notions. Accordingly, we shall quote the entire
beginning of his report [8] []{see [23, p. 261]}.
Actually, Bienayms article had not contained other applications of the method that he
recommended, but his reasoning was completely in keeping with Chebyshevs own train of
thought. This was the reason why the latter, being prepared for the solution of the general
problem posed by Bienaym by his previous research and perceiving in its solution a sure
path to proving the limit theorem, continued his deliberations in this direction. As a result, he
discovered his celebrated {integral} inequalities [8]. Let
co =
f(x)dx, c1 =
xmf(x)dx
xf(x)dx, , cm =
so that formally
f ( x )dx co c1
= + 2 + ... + mm+1 + ...
If (z)/ (z) is one of the convergents obtained by expanding the left side of (11) into a
continued fraction, and if z1, z2, , zl, zl+1, , zn, , zm are the roots of the equation (of
degree m) (z) = 0 arranged in increasing order, then, each time when f(x) remains positive
between boundaries A and B,
i = l +1
( zi )
< f(x)dx <
( zi ) z
i =l
and the boundaries in these inequalities cannot be brought closer to each other.
In short, the inequalities (12) represent the fact that for all possible positive functions f(x),
for which the expansion (11) into a continued fraction has the same convergent (z)/ (z)
(i.e., the same moments co, , cm), the extreme values of the integral in (12) correspond to
the case in which the continued fraction (11) is finite.
8. This fundamental theorem shows that, having established inequalities (12), Chebyshev
found a general method for solving the main problem of the theory of probability: To what
extent is the law of probability f(x) of a random variable determined by the given appropriate
expectations of consecutive degree (or the so-called moments), which, rather than being
arbitrary, are known to satisfy some conditions?
In addition, these inequalities open the way for answering the question of whether f(x) is
determined uniquely when the expectations of all the degrees of the variable are given (the
problem of uniqueness in the theory of moments).
Indeed, it follows from (12) that a necessary and sufficient condition for this uniqueness is
that the difference between the consecutive roots zl and zl+1 of the denominators (z) (of
degree m) of the appropriate convergents tended to zero for any l. Indeed, in this, and only in
this case any real number a is the limit of zl and the difference between the extreme parts of
(12) tends to vanish. Thus, Chebyshev now found the clue to proving the limit theorem: a
deeper study of the continued fraction (8), whose most important properties he came to know
already in 1859 (6), will show that the totality of all the moments of the normal law
uniquely determines it. Then, it still remained necessary to specify the demonstration of the
fact that, when the number n of independent variables u1, u2, , un increases, the magnitudes
E(u1 + u2 + + un)l/Bnl/2,
Bn = E(u1 + u2 + + un)2 = Eu12+ Eu22+ + Eun2, Eui= 0,
have as their limits the corresponding moments of order l of the normed Laplace Gauss
normal law. In the sequel, we shall see that this fact was apparently obvious to Chebyshev
already in his young years, when he had been studying classical literature, and that it served
him as a guiding star.
However, he still had no rivals in the new fields of mathematics created by him, and he did
not hurry either with publishing the proof of the inequalities discovered in 1873 [8] or with
surmounting the last obstacles separating him from his goal contemplated long ago. Indeed,
Chebyshevs efforts during the entire next decade had been directed to the other branches of
his versatile creative work, mostly to examining and constructing mechanisms on the basis of
his theory of functions least deviating from zero, which was another aim also delayed by him
for a long time.
In the meantime, however, Markov published a paper [17a] with a general, very clever and
simple proof of the abovementioned Chebyshev inequalities. After the appearance of this
remarkable work, whose author deeply penetrated the essence of the ideas, and simplified the
methods of his teacher, Chebyshev at once proceeded to conclude his prolonged investigation
of the limit theorem for the sums of a large number of independent variables.
9. Chebyshev made a decisive step in this direction in a memoir [9] reported to the
Academy of Sciences in 1886. After some modification of the extreme parts of inequalities
(12) which followed from the identical dependences between the numerators and the
denominators of the consecutive convergents of continued fractions, Chebyshev applied here
his inequalities for estimating the boundaries of the integral
f(x) dx, f(x) > 0
f(x) dx = 1,
x2f(x) dx = (1/q2), ,
1 3...( 2m 3)
q2m 2
x2m1 f(x) dx = 0.
The transformation of inequalities (12) mentioned above leads to the conclusion that, if
two functions, f(x) 0 and f1(x) 0, satisfy conditions (13) (i.e., if they are determined by
the moments up to the (2m 1)-th order inclusively) corresponding to the normal law
exp ( q2x2/2) dx,
(q/ 2 )
then, for any m,
f1(x) dx| < 1/(To + T1 + + Tm-1)
f(x) dx
where Ti =
l(z)
(v)/(i!q i) and
= exp (q2z2/2) [d l exp (q2z2/2)/dzl]
are the abovementioned denominators of the convergents of the continued fractions emerging
exp( q 2 x 2 / 2)
dx.
Chebyshev derived the required lower boundary of the sum
Sm-1 = To + T1 + + Tm-1
which increases to infinity with an increasing m (so that the law having all the normal
moments should be identical with the normal distribution) by a curious trick. The finite
l(v)
+ q2v
l1(v)
+ (l 1) q2
( o(v) = 1) which determines the polynomials l(v) connecting the denominators of the
consecutive convergents easily provides an equation for Tl (v):
lTl (q2v2 l + 1)Tl1 + (q2v2 l + 1)Tl2 (l 2) Tl3 = 0.
Chebyshev solved this latter by introducing a generating function
(t) = To + T1t + T2t2 + Tnt n +
After simple transformations the differential equation
(1 + t) (1 t2) (t) = [q2v2 (q2v2 1)t + t2] (t)
is obtained from which
(t) = C
exp[q 2v 2t /(1 + t )]
1 t2
where C = 1 since (0) = To = 1. It immediately follows from (17) that the infinite series of
non-negative numbers (T1 + T2 + + Tn + ) diverges which means that the sum (15)
increases unboundedly as m
. This, however, was not sufficient since, as we saw, for
Chebyshev a limiting formula established for m
was less essential than its error for a
finite m, and this very fact reflects the main aim and the virtue of his inequalities (12).
It is interesting that Chebyshev connected his trick, by whose means he obtained the lower
boundary for the sum (15) and which is applicable to any power series with non-negative
coefficients, with a simple particular case of inequalities (12); and that it actually served as
the basis of his classical proof of the law of large numbers. It should also be noted that
Chebyshev introduced here, although in other notation, Stieltjes integrals:
So as to apply to the sums
Tlt l, To + T1t + + Tm1 t m1
what was there [in inequalities (12) S.B.] provided for integrals, we represent them as
integrals
Yt xdx,
Yt xdx
where Y is a function equal to zero at all values of x not adjacent to 0, 1, 2, ; and for
values of x infinitely close to 0, 1, 2, taking such values that the integrals
Ydx,
unboundedly approach To, T1, T2, as
stated, we shall have
Yt xdx =
tends to zero. For this function, defined as just
Tlt l = (t),
Yt xdx = To + T1t + + Tm1 t m1.
It is sufficient to replace Ydx by dT where T is the appropriate step-function to obtain the
Stieltjes integral. It is therefore apparent that Chebyshev regarded the introduction of these
integrals into probability theory as a technical necessity; he actually made use of a similar
tool in his investigations.
Assuming that
m 1 = [t
(t)/ (t)] + 1
Chebyshev thus arrived 6 at
Tit i (t)
t[ (t )]2
(0 t < 1)
t (t ) + (t )
for any function (t) with non-negative coefficients. This inequality cannot obviously be
improved since for
(t) = To + Tm1 t m1
it reduces to To + Tm1t m1 To. If (i) persists, inequality (18) all the more leads to
Sm1 =
Ti > (t)
Issuing from (19) and (17), Chebyshev obtained, after some transformations and
simplifications of the right side of (19),
( m 3)3 m 1
Sm1 > [2/(3 3)] 2
( m 2m + 3) ( q v + 1)3
In virtue of (14) the following theorem takes place: If function f1(x) remaining positive
satisfies (13), then
f1(x) dx (q/ 2 )
exp ( q2x2/2) dx| <
( m 2m + 3) ( q 2 v 2 + 1)3
(3 3/2)
(i.e., tends to zero when n increases not slower than a magnitude of order 1/ m).
As stated above, Chebyshev could have rightfully formulated this rigorously proved
fundamental result in the now generally accepted notation for the Stieltjes integral.
10. Chebyshev made the last step in concluding the proof of the limit theorem on sums of
independent random variables in 1887 [4]. The first of the two theorems mentioned in the
title of [4] was the proposition of 1867 (the law of large numbers) that Chebyshev, as though
summing up all his investigations in probability, reproduced here in the beginning of this
paper before proceeding to the formulation and demonstration of the second theorem.
It only remained for him to prove that, as n
E (u1 + u2 + ... + un ) 2 k
E (u1 + u2 + ... + un )2 k +1
3(2k
1),
k +1 / 2
where Eui = 0, Eui2 = ai(2), Euim = ai(m), Bn = a12 + a22 + + an2. Indeed, he believed it
obvious that if the sequence of the laws of probability Pn(x) is such that
xkdPn(x) = ck =
xkdP(x),
where the moments ck uniquely determined some law P(x), then Pn(x) P(x). Markov was
the first to prove with complete rigor this property which is really easy to establish by means
of the Chebyshev inequalities.
It should be admitted that Chebyshevs proof of the limiting equalities (21) did not entirely
comply with the demands of rigor which he himself enunciated in his youth [1; 2]: here, he
applied a method based on a Laplace transform differing from his usual methods of mature
age. Had he not hurried with the publication of this last writing which appeared only a few
months after the one considered just above [9], he would have possibly replaced the
extremely interesting heuristic transcendental method by a more rigorous algebraic
demonstration probably known to him for a long time.
Somewhat changing Chebyshevs notation, we shall describe the train of his thought s
indicating the essence of its deficiency and explaining how, without any special difficulties,
he could have removed it. He considered the function
n(s) = E exp
1 + (s2/2) +
s(u1 + u2 + ... + un )
= E exp(suk/ Bn ) =
(sm/m!) E [(u1 + u2 + + un)/ Bn ]m
m= 3
where s was an arbitrary constant. Does this expectation exist? Of course it does for purely
imaginary values of s, but Chebyshev had not specified this, and, in addition, without
providing any explanation, he expanded
E exp(suk/ Bn ) = 1 +
( m)
s 2 ak
s m ak
2 Bn
m! Bn
into a power series. This, however, will not converge at any |s| > 0. For this expansion to
become valid for |s| < 1 it would have been sufficient, say, to introduce the restriction
[|ak(m)|/Bnm/21] m!ak(2)
0 as n
(m > 2)
. Then, if |s| 1/2, it will follow from (23) that
|E exp(suk/ Bn ) 1
|< k n
|sm| < 2 n[ ak(2)/Bn]|s|3.
Taking the logarithms of (23), Chebyshev obtained a formal expansion
ln E exp(suk/ Bn ) = ln (1 +
+ ) =
3! Bn
Without condition (24) this operation is invalid; but, having introduced this restriction, we
ln (1 +
+ ) = ln[1 +
(1 +
ln[1 +
n(s)
s 2m
m( 2 Bn )m
[ak(2) (1 +
k n)]
k n)],
|s| 1/2, | k| < 2,
= (s2/2) (1 +
k n)
+ , | k| < 2.
Thus, excepting the first one, all the coefficients of this power series that converges at |s|
1/2 will tend to zero if
[ak(2)]m/Bnm} ={
[ak(2)]m/[
(ak(2))]m}
However, in virtue of (24),
ak(2m) Bnm1
ak(2) n(2m!) = (2m!)Bnm n,
or, for every given m,
{[
ak(2m)]/Bnm} < (2m!)
and, since [ak(2)]m ak(2m), (27) follows. Therefore, at |s| 1/2
lim ln
= (s2/2) as n
and, in accord with a property of Taylor series, all the coefficients of the series for
convergent at |s| 1/2, tend to the coefficients of the series
exp (s2/2) =
n(s),
[s2i/(2i)!]1 3 (2i 1)
respectively 7.
We see now that condition (24) is sufficient for making Chebyshevs reasoning quite
rigorous without adding anything essential to it. Therefore, it should be considered in justice
that he had in essence completely proved the limit theorem on sums of independent variables
under the {additional} condition (24). This restriction covers all the practically important
cases. Indeed, it is fulfilled if 8
|uk|/ Bn <
since then, if m > 2,
|ak(m)| ak(2)(
Bn )m2
{|ak(m)|/Bnm/21}
ak(2) < m! ak(2) n.
Regrettably, Chebyshev had not introduced any restriction similar to (24) and the limit
theorem in his excessively general form was wrong.
In 1898 Markov [17b] proved by direct algebraic calculation that the limiting equalities
(21) were valid if only
{|ak(m)|/Bnm/2}
for all integer values of m > 2. This condition is somewhat weaker than restriction (24), and
Markov had thus filled in the technical gap in Chebyshevs brilliant proof based on the
method of moments.
11. Markov was undoubtedly the most outstanding exponent of Chebyshevs direction of,
and ideas in probability theory. He was the closest to his teacher in the nature and the
sharpness of his mathematical gift. Whereas Chebyshev, especially towards the end of his
life, as well as in his lectures, sometimes deviated from his own demands of clear
formulations and rigor of proof in probability, Markovs classical course in the calculus of
probability and his original memoirs were specimens of preciseness and lucidity of
exposition 9, which contributed to the maximal extent to the transformation of probability
theory into one of the most perfect branches of mathematics and to the wide dissemination of
Chebyshevs direction and methods. As we shall see below, the later deep analysis of
dependences between observed random phenomena made by Markov in the Chebyshev spirit
enabled him to extend essentially the domain of probability by introducing into consideration
dependent random variables.
My aim does not include the reviewing of all the modern achievements in probability.
These would have been impossible without the robust mathematical base of the theory
constructed by Chebyshev and consolidated by Markov. I would only like to indicate the
stages of the subsequent history of the problems connected with the LLN and the limit
theorem, which, all of them, directly adjoin Chebyshevs investigations and develop his
Let us first dwell on the LLN. Markovs simple remark [17d] directly following from
0 where Bn = E(u1 + u2 + + un)2
Chebyshevs inequality [3] that the restriction (Bn/n2)
is sufficient for the applicability of the law to variables u1, u2, , un (Eui = 0), led him, and,
later, other authors, to a number of interesting sufficient conditions for the same fact both for
independent and dependent variables ui. And, by introducing supplementary variables ui
such that ui = ui when |ui| Ln, i n, and ui = 0 if |ui| > Ln, Markov became able to prove
the applicability of the same law in a number of other cases in which Eui2 (and therefore Bn)
do not exist. For example, he generalized the Chebyshev theorem [3] in the following way: If
for some p > 1 E|ui| p < L where L is some constant, then the law is applicable to these
independent variables ui.
Among other developments obtained in a similar way, we indicate a remarkable in its
simplicity and maximal (in some sense) generality theorem due to Khinchin [21b]: If all the
independent variables ui obey one and the same law of probability and Eui exist, then the
LLN is applicable to ui. The derivation of more or less general necessary conditions for the
applicability of this law to independent variables is also based on the Chebyshev inequality
and on a simple remark [11d] that the law cannot be applied to variables ui in the particular
case in which their laws of probability are symmetric and such that there exists a positive
number c for which the probability that |ui| cn at least for one i (i n) does not tend to zero
as n increases.
The Kolmogorov theorem [13] based on an appropriate development of Chebyshevs
classical reasoning essentially supplements Chebyshevs inequality: The probability that all
the inequalities
|u1 + u2 + + ui| t Bn ,i = 1, 2, , n
for independent variables ui are valid is higher than 1 1/t2. Bernstein [11g] generalized
these inequalities onto dependent variables ui having Eui = 0 for any values of u1, u2, , ui1.
The determination of a more precise higher boundary than (1 1/t2) for the probability in
the Chebyshev inequality
|u1 + u2 + + ui| t Bn ,Bn = c1(2) + c2(2) + + cn(2)
irrespective of n necessarily demands that some more or less essential restrictions be
introduced. We [11b] note, for example, the estimate
Q < exp ( t2)
of the probability Q that
u1 + u2 + + un 2 t Bn .
It is valid for all values of n and t > 0 if t [ Bn /(2H)] under the condition that
|Eukm| < (1/2)ck(2) Hm2m!
where H is some constant. The proof of this proposition is also some development of
Chebyshevs classical reasoning. In addition, the same estimate is obtained for dependent
variables provided that Eui = 0 and Eui2 = ci(2) for any previous value of uk (k < i);
furthermore, this estimate persists [11g] when the inequality (30) is replaced by the
appropriate inequalities as in the Kolmogorov theorem.
12. The work of Chebyshev was of no lesser fundamental importance for the further study
of the conditions for the application of the limit theorem. It is necessary to dwell here first of
all on the case in which the variables ui are independent. We have already discussed
Markovs specifications of Chebyshevs proof based on the method of moments that the
former introduced and thus made the latters reasoning irreproachably rigorous by stipulating
that the variables ui not only possess moments, i.e., expectations of any order, but in addition
satisfy the demand (29). Markov [17c] obtained this result in 1898.
Soon after that, in 1901, Liapunov [16a; 16b] remarkably weakened the restriction (29).
He also was one of Chebyshevs closest students, and he experienced the powerful influence
of his teacher. It is known, for example, that it was Chebyshev who suggested to him the
problem of the figures of equilibrium of a rotating liquid that came to occupy a central place
in his studies; incidentally, this fact testifies that Chebyshevs scientific interests went
beyond those branches of mathematics where his personal original creative work had
manifested itself. Nevertheless, Chebyshevs influence upon Liapunov, who, with respect to
the power of his gift, was second to none either in Russia or in the West, was not really
exceptional. Liapunov, better than the other representatives of the Petersburg school,
understood, and was able to appreciate the achievements of the West-European
mathematicians of the second half of the last {of the 19th} century 10 who had outlined
precise boundaries for the methods of classical transcendental analysis, and made them no
less reliable than Chebyshevs algebraic methods. This very fact explains why Liapunov
approached Chebyshevs problems more independently than his other students. Whereas
Markov perfected the methods of his great teacher and applied them to new problems,
Liapunov, when thinking about the essence of the limit theorem, understood that the method
of moments did not facilitate the problem and only shifted its main difficulty elsewhere.
Indeed, the Laplace transform, applied by Chebyshev in his last memoir of 1887, consisted in
considering the expectation
Eesx = (s) =
esxf(x) dx
for an arbitrary value of the parameter s. Instead of the totality of moments cm = Exm, the
(s) =
cmsm/m!
can itself characterize the density f(x) or the integral law of probability
if only the series (22bis) converges, which, as we saw, is also an essential condition for the
correctness of the relevant portion of Chebyshevs reasoning.
However, it is not at all necessary to use the expansion (22bis) into a power series; there is
even no need for supposing that the moments ck exist provided we shall only assign purely
imaginary values to s, s = it, since the integral
eitxf(x) dx=
(t) = (it) =
eitxdF(x)
exists for all real values of t.
The function (t) which is now called the characteristic function of the law of distribution
of variable x, uniquely determines the law F(x) irrespective of the existence of ck. In addition,
when considering the case of
Xn = 1 / Bn (u1 + u2 + + un)
Liapunov proved that, for the characteristic function of the law of probability of Xn to
converge to
exp ( t /2) = (1/ 2 )
eitx exp ( x2/2) dx,
i.e., to the characteristic function of the normed normal law
(1/ 2 )
exp ( x2/2) dx,
it is sufficient that, at least for one > 0,
[ak(2+ )]/Bn1+
} = (1/Mn, )
0, ak(2+ ) = E| uk|2+ .
This condition is thus sufficient for the applicability of the limit theorem to suns of
independent variables ui. Following Chebyshevs precept, Liapunov also indicated the higher
boundary of the error of the limit theorem. For example, if = 1, the order of this error is (ln
Mn, 1)/ Mn, 1.
So, Liapunov established with perfect rigor and by an absolutely different method the limit
theorem for sums of independent variables under a much more general condition than
Markov. The latter, however, became able to exonerate the method of moments.
Supplementing it by a short reasoning similar to that, which he had applied for generalizing
the conditions for the applicability of the LLN, he fully demonstrated the limit theorem under
the general Liapunov condition.
Later, in 1926, when applying in essence the same remark as Markov did, I [11c] provided
a still more general condition for the application of the limit theorem 11 that also covered
such cases in which Eui2 did not exist. For example, if the variables ui might take values
m, where m is any positive integer, with probabilities pm = 3/( 2m2), so that Eui = 0 and
Eui2 = (6/ 2)
(1/m) = ,
m =1
then the probability
P[(to/ ) 6n ln n <
ui < (t1/ ) 6n ln n ]
exp ( t2/2) dt.
In connection with this illustration it should be noted that the meaning of the limit
theorem, depending on whether some or other conditions are satisfied, is not quite the same.
Whereas the very method of proof applied by Chebyshev and Markov testified that, under the
latters conditions, all the moments of the sum (ii) tended to the normed normal moments (for
example, limEXn4 = 3 as n
), the Liapunov demonstration does not show it at all; and,
possibly does not exist at > ,
E|Xn|2+ will also be
since under condition (32) E|ui|
meaningless. It can be proved [11h] that the Liapunov condition with a given > 0 is
necessary and sufficient for E|Xn|p with p 2 + to have the corresponding normal moment
as its limit. In the example just above the Liapunov condition is not fulfilled for any > 0
and we have here, as n
| u1 + u2 + ... + un |
lim E {
6n ln n
} p = (1/ 2 )
|x|pexp ( x2/2) dx
only for p < 2. The Norwegian mathematician Lindeberg [15] derived a condition sufficient
for EXn2
1; here, p = 2.
In general, during the last twenty years a large number of studies have thrown light, from
various points of view, on some important properties of the normal law and on the conditions
for its applicability to sums of independent variables. Nevertheless, their findings are far less
fundamental than the result attained by Chebyshev and his students, Markov and Liapunov.
It should be noted, however, that during the same period a number of absolutely new
problems have been formulated and solved. The most interesting in its essence was the
discovery of the general form of the limiting law for the sums of independent variables in
those exceptional cases in which it is not normal. The pertinent subtle investigations, the
most important of which were due to Kolmogorov, Khinchin and Lvy, were based on the
appropriate use of the Liapunov method of characteristic functions. They demanded a deep
study of the properties of these functions and revealed an unexpected fact: The so-called
Poisson law is in some sense even more universal than the Laplacean normal distribution
[21c]. It is curious that the Petersburg mathematicians had completely overlooked the
Poisson law which statisticians call the law of small numbers 12 and which also plays a
considerable part in physical applications of probability.
13. Turning now to the generalization of the main laws of the theory of probability onto
dependent variables, we recall that Chebyshev himself did not study this subject; however,
since Markov was the founder of this central field of modern probability, and consistently
developed Chebyshevs ideas in general and the method of moments in particular, we must
briefly discuss this point. Following Chebyshev in that he believed that each new general
domain of research should be approached by considering appropriate typical problems
having a clearly seen and real meaning and admitting of a precise mathematical formulation,
Markov displayed here his deep flair of an outstanding natural scientist and directed his
attention to a class of dependent random variables exceptionally important for applications.
He called his objects chains, and their present-day scientific designation is Markov chains 13.
The British biologist and statistician Galton was the first to consider such dependences
under some particular assumptions. Drawing on rather extensive experimental materials, he
attempted to explicate the Darwinian theory of heredity in a mathematical form. However,
we do not find any satisfactory mathematical investigation of the properties of stochastic
patterns similar to the Markov chains either in his works, or in the writings of physicists who
came to study them almost at the same time as Markov did. The closeness of Markovs ideas
to the notions that originated simultaneously in various fields of natural sciences testifies to
their vitality and methodological importance. Indeed, a Markov chain is a stochastic
transformation of a usual deterministic process; it is characterized by the fact that its dynamic
state at a given moment fully determines its further course irrespective of all the previous
states. More precisely, Markov defined a chain as a sequence of such random variables (or
events, or states) X1, X2, , Xn, that, after Xn takes some definite value, the probability
that the next variable Xn+1 takes one or another value becomes fully determined and does not
depend on the values of the previous variables Xi, i < n. Excluding the case in which the
chain unboundedly approaches a determinate process, Markov showed 14 that the dependence
between two links of the chain, Xi and Xi+h, sufficiently far from one another, weakens so
rapidly that , as n increases, the law of probability of Xn tended to a law independent from the
initial state X1. Under some very general assumptions, this important theorem might justify
the hypothesis of a uniform distribution of probabilities at set conditions that forms the basis
of various physical theories. By generalizing, as indicated above, the Chebyshev inequality
and under the same restrictions Markov also derived an extension of the LLN as a corollary
to the fact that the dependence between remote links of a chain was very weak.
The problem of the limiting applicability of the normal law to the sum of terms
constituting a chain presented more considerable difficulties. The method of moments
demanded the calculation of the expectations
( X i / Bn
for all positive integer values of m and a proof that they tended to the corresponding normal
moments. In the simplest cases the method of generating functions led to this goal, but the
algebraic method similar to that which Markov had applied in 1898 to independent variables,
proved here very unwieldy and Markov derived the demanded result under rather general
conditions only owing to his mastery.
Later on I had generalized his findings basing my work on the Liapunov method of
characteristic functions. This, however, was not here directly applicable: the advantage of
these functions consisting in that, for independent terms, a characteristic function of a sum of
two variables is equal to the product of such functions for each of these terms, falls away for
the case of dependent variables. It was therefore necessary to supplement this method by the
idea of separating chains into sections in such a way, that, after eliminating a number of
intermediate links, the limiting law of probability for the entire sum would not differ from
that of the remaining sum consisting of sufficiently weakly connected one with another, and
in some precisely defined sense almost independent groups of terms. This idea is apparently
applicable not only to Markov chains, but also to more general classes of dependent
variables. The most difficult and interesting is the study of the case in which the chain is
singular, i.e., very close to a strictly deterministic process. The discussion of the findings
derived in this direction is beyond the boundaries of this paper. I only note that this approach
and many useful supplementary pertinent lemmas directly adjoin the problems examined by
Chebyshev. Thus [11f], the following theorem is in his spirit: For any Markov chain the
variance B(Sn) of the sum of variables X1, X2, , Xn, Sn = X1+ X2 + + Xn, satisfies the
B(Sn)
2 [n/2]
where h is the mean conditional variance of Xh after the values of Xh1 and Xh+1 become
known. The boundary is actually attained when X1, X3, are determinate and X2, X4, are
independent. The problem about the limiting law of Sn for the case in which it differs from
the Laplace Gauss distribution is of a considerable theoretical and practical interest. The
appropriate contributions are mostly closely connected with Markov chains, but the field in
question also makes use of a number of new ideas and is too extensive for being discussed
I am concluding this brief review of the development of Chebyshevs ideas in which, his
celebrated students, Markov and Liapunov, participated above all. We see that the theory of
probability owes to the work of Chebyshev and his school its present-day maturity that
ensures its reliable application to most diverse real phenomena. The genius of Chebyshev and
his associates, who, in this field, left West-European mathematicians far behind, overcame
the crisis of probability that arrested its development a hundred years ago.
1. {Bernstein mentioned De Moivre (see above) but had not discussed his finding; and
Laplaces work on the CLN did not date back to the 18th century.}
2. {Bernstein overlooked two other fields of application of the early probability theory,
viz., population statistics and the treatment of observations.}
3. {See Note 7 to Gnedenkos essay on Ostrogradsky in this book.}
4. Its subtitle is Extrait dun mmoire russe sur lanalyse lmentaire de la thorie des
probabilits, and, although the main part of [2] is lacking in the Dissertation [1], this is
obviously a reference to [1]. It might be concluded that [2] had existed in some form when
Chebyshev defended his dissertation in 1846 in Moscow. S.B. {A later commentary on [1] is
Prokhorov (1986, 7).}
5. It should have been mentioned that the trials were independent. Actually, here and in all
his works in general, Chebyshev only considered independent events. Poissons essential
mistake, which Chebyshev indeed corrected, consisted in that he had indiscriminately applied
his theorem to any events.
6. Inequality (18) can be easily obtained directly by noting that
Tit i =
Tit i [1/(m 1)]
i=m
iTit i
[t (t )]
whence
= (t)
7. {The equality just above is obviously wrong.}
8. It was subsequently shown that the most general form of the limit theorem can be easily
derived when issuing from condition (28) [11c].
9. {Students hardly liked Markovs treatise [17d]; in 1910, he himself (Ondar 1977, p. 21)
owned that he often heard that his presentation of the MLSq was not sufficiently clear.
Later editions were not better. And the opening lines of Markovs memoir [17e] present a
specimen of unreadable phrases.}
10. {Nevertheless, Liapunov (1895, pp. 19 20) called Riemanns ideas extremely
abstract and his investigations pseudo-geometric, having nothing in common with
Lobachevskys deep geometric studies. Then, Markov hid his deep flair of an outstanding
natural scientist: he did not even hint at the possibility of applying his chains in natural
sciences. A possible explanation of his attitude is found in his letter of 1910 to Chuprov
(Ondar 1977, p. 52): I shall not go a step out of that region where my competence is beyond
any doubt. I also note that it was Bernstein himself [11c, 16] who introduced the term
Markov chain. Finally, Bernstein had not elaborated his remark (below) about the prehistory
of chains. I myself (Arch. Hist. Ex. Sci., vol. 39, 1989, pp. 364 365) listed several pertinent
developments (but did not mention Galton).}
11. Feller [20] proved that this condition was also necessary under the natural assumption
that each term was negligible as compared with the entire sum.
12. {Bernstein overlooked Markovs studies of the stability of statistical series. They were
partly connected with the law of small numbers; hence, with the Poisson law.}
13. {It was Bernstein himself [11c, 16} who introduced the term Markov chains. Markov
hid his deep flair of an outstanding natural scientist: he did not even hint at the possibility of
applying his chains in natural sciences. A partial explanation of his attitude is found in his
letter of 1910 to Chuprov (Ondar 1977, p. 52): I shall not go a step out of that region where
my competence is beyond any doubt. Finally, concerning the prehistory of Markov chains
(see below): Bernstein regrettably did not elaborate his remark about Galton. I myself (Arch.
Hist. Ex. Sci., vol. 39, 1989, pp. 364 365) listed several facts pertaining to that prehistory
(but did not mention Galton).}
14. Almost at the same time Poincar [18] independently arrived at the same result in some
particular cases.
P.L. Chebyshev
1. (1845), (
(Essay on an
Elementary Analysis of the Theory of Probability). Reprinted: '
2. (1846), Dmonstration lmentaire dune proposition gnrale de la thorie des
probabilits. J. reine u. angew. Math., Bd. 33, pp. 259 267.
3. (1867), Des valeurs moyennes. J. math. pures et appl., t. 12, pp. 177 184.
4. (1887, in Russian), Sur deux thormes relatifs aux probabilits. Acta math., t. 14, 1890
5. (1936),
(Theory of Probability). Lectures read 1879/1880 as
taken down by A.M. Liapunov. M. L. Translated into English: Berlin, 2004.
6. (1855, in Russian), Sur les fractions continues. J. math. pures et appl., t. 2, 1858, pp.
7. (1859), Sur le dveloppement des fonctions une seule variable. Bull. Cl. phys.-math.
AN Psb, t. 1, pp. 193 200.
8. (1874), Sur les valeurs limites des intgrales. J. math. pures et appl., t. 19, pp. 157
9. (1887, in Russian), Sur les rsidus intgraux qui donnent des valeurs approches des
intgrales. Acta math., t. 12, 1888 1889, pp. 287 322.
(1899 1907), uvres, tt. 1 2. Psb., in Russian and French, contain contributions [2; 3;
6; 7; 9] in t. 1 and [4; 8] in t. 2. French edition reprinted (New York, 1962).
10. Bernoulli, J. (1713, in Latin), Wahrscheinlichkeitsrechnung. Frankfurt/Main, 1999.
11a. Bernstein, S.N. (1922), Sur le thorme limite du calcul des probabilits. Math. Ann.,
Bd. 85, pp. 237 241.
11b. --- (1924) On a modification of the Chebyshev inequality and on the error of the
Laplace formula.
, vol. 4 (Coll. Works). N.p., 1964, pp. 71 79. (R)
11c. --- (1926), Sur lextension du thorme limite du calcul des probabilits aux sommes
de quantits dpendantes. Math. Ann., Bd. 97, pp. 1 59.
11d. --- (1946),
(Theory of Probability), 4th edition. M. L. First
edition, 1927.
11e. --- (1928), Addition larticle Sur les sommes de quantits dpendantes. Doklady AN
SU, pp. 55 60.
11f. --- (1936), Dtermination dune limite infrieure de la dispersion des sommes des
grandeurs lies en chaine singulire. MS, vol. 1, pp. 29 37.
11g. --- (1937), On some modifications of the Chebyshev inequality. Doklady AN SU, vol.
17, pp. 275 277. (R)
11h. --- (1939), Several remarks on the Liapunov limit theorem. Ibidem, vol. 24, pp. 3 7.
12. Bienaym, I.J. (1853), Considrations lappui de la dcouverte de Laplace, etc. C.r.
Acad. Sci. Paris, t. 37, pp. 309 324. Reprinted 1867.
13. Kolmogorov, A.N. (1928), ber die Summen durch den Zufall bestimmten
unabhngigen Grossen. Math. Ann., Bd. 99, pp. 309 319.
14. Laplace, P.S. (1812), Thorie analytique des probabilits. Oeuvr. Compl., t. 7. Paris,
15. Lindeberg, J.W. (1922), Eine neue Herleitung des Exponentialgesetzes, etc. Math. Z.,
16a. Liapunov, A.M. (1900), Sur une proposition de la thorie des probabilits. Izvestia
(Bull.) AN Psb, sr. 5, t. 13, No. 4, pp. 359 386.
16b. --- (1901), Nouvelle forme du thorme sur la limite des probabilits. Zapiski (Mm.)
AN Psb, sr. 8, cl. phys.-math., t. 12, No. 5, pp. 24.
17a. Markov, A.A. (1884), The proof of some of Chebyshevs inequalities.
(Sel. Works), M. L., 1948, pp. 15 24. (R)
17b. --- (1899), The law of large numbers and the method of least squares. In Markov
(1951, pp. 231 251). Translated in this book.
17c. --- (1898), Sur les racines de lquation Izvestia (Bull.) AN Psb, sr. 5, t. 9, pp. 435
(Calculus of Probability). Psb. Later editions:
1908, 1913, 1924. German translation: Leipzig Berlin, 1912.
17e. --- (1906), Extension of the law of large numbers, etc. Translated in this book.
17f. --- (1908), Extension of the limit theorem, etc. Translated in this book.
17g. --- (1910), Investigation of the general case of trials, etc. Translated in this book.
17h. --- (1911), On a case of trials connected into a generalized chain. In Markov (1951,
pp. 417 436).
17i. --- (1912), On trials connected into a chain by non-observed events. Ibidem, pp. 437
17j. --- (1907, in Russian), Recherches sur un cas remarquable dpreuves dpendantes.
Acta math., t. 33, 1910, pp. 87 104.
17k. --- (1915), Application of the method of expectations to connected series of
magnitudes. Izvestia (Bull.) AN Psb, sr. 6, t. 9, pp. 1453 1484. (R)
Markov, A.A. (1951),
(Sel. Works). N.p.
18. Poincar, H. (1912), Calcul des probabilits. Paris.
19. Poisson, S.-D. (1837), Recherches sur la probabilit des jugements, etc. Paris.
20. Feller, W. (1935 1937), ber den zentralen Grenzwertsatz der
Wahrscheinlichkeitsrechnung. Math. Z., Bd. 40, pp. 521 559; Bd. 42, pp. 301 312.
21a. Khinchin, A.Ya. (1928), Sur la loi forte des grands nombres. C.r. Acad. Sci. Paris, t.
186, pp. 285 287.
21b. --- (1929), Sur la loi des grands nombres. Ibidem, t. 188, pp. 477 479.
21c. --- (1938), '
(Limit
Laws for Sums of Independent Random Variables). M.
Liapunov, A.M. (1895), Chebyshev. In Chebyshev, P.L.,
(Sel. Math. Works). M. L., 1946, pp. 9 21. (R)
Ondar, Kh.O., Editor (1977, in Russian), Correspondence between Markov and Chuprov,
etc. New York, 1981.
Prokhorov, Yu.V. (1986), The law of large numbers and the estimation of the probabilities of
large deviations. In Bernoulli, J. (
(On the Law of Large Numbers).
Editor, Yu.V. Prokhorov. M., pp. 116 150. (R)
6. N.A. Usov. A Remark concerning the Chebyshev Theorem
. Matematich. Sbornik, vol. 2, 1867, pp. 93 95 of second paging
The author was Ostrogradskys student (Dobrovolsky & Kiro 1967, p. 125). He published
his note immediately after Chebyshevs pertinent contribution had appeared (and he referred
to its Russian version which was issued at the same time as the French version mentioned in
the References). I have insignificantly changed Usovs text by replacing, where possible,
integrals, in accord with his notation, by a or a1.
Usovs attempt was forgotten; the justification of the arithmetic mean and of the MLSq in
general by a reference to Chebyshev (1867) is usually ascribed to Yarochenko (1893a;
1893b) although Maievsky had preceded the latter. Markov, however, reasonably rejected
such efforts, see my paper in this book where I cite Markovs negative and reasonable
opinion about such efforts.
That the treatment of observations had been highly topical is once again proved by
Chebyshevs remark (1887) accompanying his proof of the CLT. That proposition, as he
stated, led to the MLSq, but only, as I add, in the Laplacean sense of treating a large
number of observations. And the Gauss mature substantiation of the method did not depend
on any law of error.
A rather remarkable corollary easily follows from Chebyshevs theorem (1867). The
principle of the arithmetic mean is known to be admitted as a postulatum for the derivation of
the type of the function ( ) of the probability of error . The theorem of Mr. Chebyshev
provides a rigorous demonstration of that principle.
Suppose that N observations were made to determine some magnitude u and resulted in
A1, A2, , An.
Denoting in Mr. Chebyshevs theorem
x = u A1, y = u A2, z = u A3,
we see that all the possible values of x, y, z, will be situated between some boundaries, call
them L and M. According to the definition assumed by Mr. Chebyshev, the expectations of
x, y, z, will be, respectively,
( )d
a=b=c=
and those of x2, y2, z2, ,
a1 = b1 = c1 =
where ( ) d is the probability that the error of observation is situated between and ( +
d ).
Then, in accord with the theorem of Mr. Chebyshev, the boundaries for the magnitude
u Ai/N
( ) d (1/t)
a1 a 2
and the probability P that this magnitude will not exceed those limits are, for any t,
P > 1 t2/N
and, consequently, for any t, it will tend to 1 as N increases.
If positive and negative errors are equally possible, then |L| = M, ( ) = ( ) and
( )d =0
and we may conclude: The probability(2) that the magnitude (1) differs from zero not more
than by
(1/t)
2 ( )d
tends to unity as N increases.
It is easy to see that a1 is the mean of the squares of the errors. Assuming that
( ) = (h/
) exp( h2 2)
we can obtain
a1 = 1/(2h2).
I think that this remark will be useful for teaching.
Chebyshev, P.L. (1867), Des valeurs moyennes. J. math. pures et appl., t. 12, pp. 177
--- (1887, in Russian), Sur deux thormes relatifs aux probabilits. Acta math., t. 14, 1890
Dobrovolsky, V.A., Kiro, S.N. (1967), Teaching of mathematics in special technological
and military schools. In
(History of National
Mathematics), vol. 2. Editor, I.Z. Stokalo. Kiev, 1967, pp. 119 126. (R)
Yarochenko, S.P. (1893a), On the theory of the method of least squares. Zapiski Novoross.
Univ., vol. 58, pp. 193 208 of second paging. (R)
--- (1893b), Sur la mthode des moindres carrs. Bull. sciences math., sr. 2, t. 17, pp. 113
7. S.N. Bernstein. The Petersburg School of the Theory of Probability
Uchenye Zapiski Leningradsk. Gosudarstven. Univ., No. 55 (ser. math. sci., No. 10),
Report read at the plenary meeting of the scientific conference devoted to the 120-th
anniversary of Leningrad University
This contribution, also valuable, obviously served Bernstein as a point of departure for his
later essay on Chebyshev (included in this book). Most of my critical historical notes inserted
there are relevant here also. Even here the author passed over in silence the alienation of the
Petersburg school from the then powerful and fruitful European direction towards abstract
structures, cf. my Note 10 to his essay.
[1] Some of the most illustrious pages of the past that we may proudly recall at the
University celebrations are the scientific achievements of the Petersburg school of the theory
of probability. The aim of my address is to throw light on the leading part, and the
importance of the works of the three main representatives of this school, Academicians
Pafnuty Lvovich Chebyshev, Aleksandr Mikhailovich Liapunov and Andrei Andreevich
Markov, who laid a firm foundation of the modern theory of probability and its applications
to scientific natural sciences.
Indeed, at the time of its origin probability theory had little in common with the general
progress of these sciences, and games of chance provided the only experimental basis for the
development and specification of its most important notions and principles. It was on these
grounds that Jakob Bernoulli, more than two hundred years ago, had discovered his
celebrated theorem that offers a clue to the understanding of the origin of regularities
inherent in mass independent individual chances, and represents the first rigorously proved,
even if a very particular formulation of the LLN.
Laplaces classical research, and, in the first place, the widely known De Moivre Laplace
limit theorem, constituted the next most important stage in the development of the theory of
probability. This proposition establishes the limiting law of distribution for the deviations of
the number of occurrences of some random event from its expectation when the trials
correspond to one and the the same elementary Bernoullian pattern and are repeated many
times over. The LLN and the normal law of distribution, or the so-called Gaussian law of
random errors, that generalizes the Laplace theorem {?}, are fundamentally important for our
Weltanschauung. They were thus discovered in their most simple form already in the 18th
century.
Another of Laplaces prominent merits consisted in that he foresaw the universality of
these laws and was the first to lead the theory of probability out from the narrow range of
games of chance onto the wide arena of scientific natural sciences. Because of his influence,
the first half of the last {the19th} century was marked by a heightened interest to, and
enthusiasm for probability theory, but many of its generalizations and applications were
insufficiently substantiated, and some of them, even those supported by Laplace himself and
Poisson, were so manifestly wrong, that later on John Stuart Mill quite deservedly qualified
them as a real opprobrium of mathematics.
Because of these failures experienced by the theory of probability in its new field,
enthusiasm gave way to disappointment and scepticism. Rare exceptions excluding, West
European mathematicians of the second half of the last {of the 19th} century believed that the
theory was nothing but a peculiar mathematical amusement that did not permit important
scientifically substantiated applications and barely deserved any serious study.
[2] However, at that time the Petersburg school of the theory of probability appeared on
the stage of the world science. Instead of the scepticism, understandable owing to the
previous mistakes, it was able to elicit some indications for a radical revision of the theorys
classical ideas and methods and for their adaptation to a profound analysis of random
phenomena in all their actual variety. Specifying the formulation of the problems of
probability theory, Petersburg mathematicians brilliantly surmounted the obstacles that had
checked its development. In particular, they solved, in a measure sufficient for all its
practical applications, both of the main contemporaneous problems, viz., the discovery of the
conditions for applying the LLN, and of the normal law.
It might be assumed that the Petersburg school had originated in 1837: from then onward,
the theory of probability became a discipline continually taught at Petersburg University.
Until 1850, its first lecturer had been Professor Vikenty Aleksandrovich Ankudovich among
whose listeners was the future head of the Petersburg school, Chebyshev. From 1850 to
1860, the course was read by Academician Victor Yakovlevich Buniakovsky, who, like his
predecessor, had been developing the ideas of Laplace and Poisson, and, attaching great
importance to the applications of the theory, largely promoted interest in probability among
the Russian public. The flourishing of the theory of probability at Petersburg University
began in 1860, when the reading of the course passed to Chebyshev. In 1882, Markov, one of
the two of his celebrated students, replaced him, whereas the second one, Liapunov, never
delivered any courses in probability and devoted to this discipline not more than two or three
of his contributions 1 having, however, utmost importance. Markov continued to teach until
the end of his days, and his treatise, matchless both in precision and clearness of exposition 2,
was no less influential than his original works in transforming the theory of probability into a
rigorous mathematical discipline.
In 1922, with the death of Markov, the Petersburg school had lost its last eminent
representative and actually ceased to exist, but its ideas spread the world over and left an
ineffaceable trace in science. Petersburg mathemati-cians generalized the principles of the
theory of probability so as to bring the widest possible classes of random events and
variables within reach of mathematical analysis. Unlike their predecessors, they formulated
the assumptions of their propositions with extraordinary preciseness and only attempted to
apply the concept of probability to such phenomena whose occurrence or non-occurrence
could be experimentally checked. Thus, for them, the LLN as well as the limit theorems
made sense not as abstract characteristics of fictitious infinite collectives, but as
approximations to quantitative relations observed in sufficiently numerous and actually
existing aggregates of some random elements.
[3] This point of view, which puts in the foreground the establishment of the appropriate
inequalities for the finite case, and the estimation of the errors of limit theorems, is already
outlined in Chebyshevs first work of 1846. There, while offering a new proof of the
generally known Poisson theorem, the author justified his research by stating that
Regardless of the cleverness of the method used by the illustrious geometer, it does not
provide the boundary of the possible error.
Filling in this gap, Chebyshev offered a very precise estimate of the studied error and
became able to ascertain the approximation furnished by the Poisson formula when applied
to a given and sufficiently long but finite series of trials. The same tendency characterizes all
the later investigations accomplished by Chebyshev; in particular, it marks his discovery,
made twenty years later, of the form of the LLN remarkable for its generality and embracing
all the practically met with totalities of independent random variables. The strikingly simple
idea of his proof {of this form of the law} is based on an inequality that states that if the
variance of a random variable is sufficiently small, its considerable fluctuations are very
The {Bienaym }Chebyshev inequality is extremely general so that the estimate of the
probability that it provides is often insufficient for practical applications. However, the
authors of all of its subsequent specifications, obtained at the expense of appropriate
restrictions, have essentially applied Chebyshevs idea. Just the same, the later
generalizations of the applicability of the LLN to totalities of either independent or
dependent variables, which provide an exhaustive solution of the problem concerning this
law, and which were accomplished mostly by Markov, and, in part, by later authors, were
based on one or another development of Chebyshevs reasoning.
The same should be said about the proofs of the so-called strong LLN which was beyond
the scope of the Petersburg school, and which, like its ordinary counterpart, is applicable to
all practically met with totalities of independent random variables. As Academician
Kolmogorov has recently shown, it is indeed applicable to all independent variables
satisfying the Chebyshev condition of restricted variations. It should be noted in this
connection that the generally accepted shortened formulation of the strong LLN is a
proposition concerning an infinite repetition of independent trials. These are actually
unrealizable, do not admit of experimental checks, and are therefore inadmissible from the
point of view of the Petersburg school. Nevertheless, the really concrete substance of the
strong law is expressed with perfect preciseness by means of the inequalities of the
Petersburg school, and its proof is based on exactly this fact.
For example, if we, for the sake of simplicity, restrict our attention to the repetition of
independent Bernoulli trials with p being the probability of the occurrence of event A, and pn
= m/n, the frequency of its occurrence in n trials, the strong LLN (in this case, the strong
Bernoulli theorem) is reduced to the following proposition: For any given magnitudes > 0
and > 0 it is possible to indicate such a sufficiently large number no, that the probability of
the simultaneous realization of the inequality (pn p) < for all values of n, no n < n + k,
will be greater than 1 however large are the values of k > 0. In this form, the only one
suitable for applications, the strong LLN does not therefore differ in essence from the
classical LLN; it is only some supplement of the latter. As already noted above, all the
recently made relevant theoretical investigations, at least insofar as they can be applied to
natural sciences, were essentially based on Chebyshevs ideas.
Among the two problems mentioned above that gave rise, in the last {the 19th} century, to
the crisis of probability theory, the one connected with the LLN was solved in the most
general way, and by essentially simpler mathematical means than those {earlier} applied by
the classics for obtaining its more or less particular solutions. This only became possible
because Chebyshev and his followers attentively and precisely analyzed the general
properties of the most diverse random variables. On the contrary, the solution of the second
problem concerning the normal law presented profound mathematical difficulties and
demanded the creation and use of new analytical methods. The complete investigation and
solution of the problem about the conditions for the applicability of the normal law in the
limiting case to the sums of very large numbers of independent random terms should
therefore be considered as the most outstanding achievement of the Petersburg school; it was
jointly accomplished by all three of its great representatives.
Chebyshev made the first essential step towards solving this problem. His way consisted in
approximately determining a monotone function representing the integral law of distribution
of a random variable, given some finite number of its successive moments, i.e., expectations
of its consecutive integral powers. Owing to his profound investigation of the extreme values
of definite integrals, very important in other problems of mathematical analysis as well,
Chebyshev inferred, in particular, that the normal law of distribution was completely
determined by the sequence of all of its moments. Using arguments that were not, however,
completely rigorous, he then ascertained that the successive moments of the sums of a very
large number of independent variables tended to normal moments, and concluded that the
limit of the distribution for the sum of such variables was the normal law. However, the two
last links of his reasoning demanded essential specification, and this was done by Markov.
[4] Here, I have to enter into some mathematical details without which it is impossible to
conceive the successive stages in the formulation and solution of the problem concerning the
normal law. Following Chebyshev, Markov only considered sums
Sn = x1 + x2 + + xn
for such independent random variables xi that possessed expectations or central moments Ci(p)
of all integer powers p. Calculating the central moments of their sum Sn, he proved, with an
indisputable mathematical rigor peculiar to all his investigations, that for {Bn = var Sn} all
integral values of p, if
Mn (p) = (1/Bn p/2)
Ci(p)
ES n
/2)t
Ci(p), n
p/2
He then ascertained that, if the moments of the sum Sn/ Bn tended to the normal moments
Ap of the corresponding orders, the distribution of this sum tended to the normal law. Thus,
basing his reasoning on Chebyshevs ideas, Markov, in 1898, for the first time discovered a
precise and sufficiently general formulation of the conditions for the applicability of the
normal law of distribution to the sums of independent random variables. These conditions
are, therefore, that, as n increases, Mn (p) tends to vanish for all integral values of p > 2. In
particular, this is fulfilled if all the magnitudes xi remain restricted, and Bn
as n
increases indefinitely, i.e., if xi/ Bn
0. Because of this, Markovs result embraced all the
former rigorously ascertained forms of the limit theorem, and, in addition, covered a
considerable part of its practical applications since the terms xi were indeed usually
restricted. Nevertheless, when schematically representing natural phenomena by a theory, the
last condition is sometimes too inconvenient. If, in such cases, one discards it, but is
reconciled with the existence of the moments of all orders, it can be difficult to verify
whether these moments obey all the Markov conditions.
The appearance, in {1900 and} 1901, of Liapunovs work was therefore an extremely
important event, both theoretically and practically. Applying quite another method, that of
characteristic functions, he formulated a remarkably simple and general condition for the
applicability of the normal law. The idea of his method goes back to Laplace, Fourier and
Dirichlet, but its rigorous realization presented, as it seemed, insurmountable mathematical
difficulties. It is Liapunovs considerable historic merit that he had overcome this obstacle
after which the method of characteristic functions secured for itself the central place in
mathematical investigations in modern probability theory. Liapunovs sufficient condition for
the applicability of the normal law to the sum Sn of independent random variables xi in the
limiting case consisted in that only one of the Markov restrictions had to be fulfilled: Mn( p)
should tend to zero at least for one value of p > 2 where p was not necessarily an integer.
Soon after Liapunov had discovered this general, and very convenient for applications
condition, Markov obtained the same result as a corollary of his own theorem by means of a
simple, but essentially important remark, which contemporary authors are using very often in
similar cases, apparently without knowing about Markovs priority. It consists in the
following: Liapunovs condition Mn( p) 0 for some p > 2 means that, although the terms xi
can be arbitrarily large, the probability of their taking values of the order of Bn is so low,
that we shall not change the limiting law of distribution for their sum by assuming that such
large values are impossible; and, as stated above, the Markov theorem will be valid for the
thus changed sum.
By referring to the same remark it was not difficult for me, in 1926, to replace the
Liapunov sufficient condition by a somewhat more general one, according to which the
individual terms xi can even posses no finite variances. In 1936, Feller, a young German
mathematician, proved that the new condition was not only sufficient, but, in a sense,
necessary for the applicability of the normal law to the sums of independent variables in the
limiting case.
One circumstance, not yet noted by anyone, should, however, be paid attention to. In a
sense rather interesting for applications, the Liapunov condition Mn( p) 0 for a given p is at
the same time sufficient and necessary. Indeed, the infinite totality of the Markov conditions
expresses not only that the distribution of Sn/ Bn has the normal law as its limit, but, in
addition, that all of its moments tend to normality since both these facts are inseparable when
the Chebyshev Markov method of moments is applied. On the contrary, the Liapunov
condition for some single p > 2, whose inalienable corollary is incidentally the fulfilment of
the pertinent conditions for all smaller values of p, does not assume anything about the
moments of the higher orders, which, generally speaking, can even be non-existent.
It can be shown, however, that the Liapunov condition of the order of p is equivalent to
E(Sn/ Bn ) p having as its limit the corresponding normal expectation; for integral and even
values of p this directly follows from Markovs reasoning. This condition for power p, or the
equivalent single demand that the moment of this order for magnitudes (Sn/ Bn ) p has as its
limit the corresponding normal moment, is therefore necessary and sufficient for the law of
distribution of the sum (Sn/ Bn ) to tend to the normal law, and, at the same time, for its
moments not higher than p to tend to the normal moments. For p = 4 this remark is also of
some practical importance, since, when fitting empirical curves of distribution, statisticians
usually make use of the moments up to and including the fourth order. It follows that the socalled excess being close to zero is, under appropriate conditions, necessary and sufficient for
the closeness of the pertinent distribution to normality. Owing to Liapunovs and Markovs
investigations, which supplemented each other, the historic merit of the complete elucidation
and final solution of the general problem about the conditions for the applicability of the
normal law to sums of independent random variables thus exclusively belongs to the
Petersburg school of the theory of probability.
[5] The last stage in the development of this school that took place during the first two
decades of our {of the 20th} century was only connected with Markov. After the problems
posed by the LLN and the normal law were solved for the case of any independent variables,
Markov, for the first time ever, precisely posed and began to extend these laws onto
dependent variables. He thus sketched the main line of the further development of probability
theory in the 20th century, i.e., the investigation of all regularities occurring when a large
number of random variables somehow connected with each other are aggregated. As stated
above, Markov, without experiencing any particular difficulties, extended Chebyshevs
reasoning onto dependent variables and offered an essentially exhaustive solution of the
problem of the LLN for this case as well. When applying this law to mass aggregates of
elements depending on each other, we need not anymore be afraid of repeating bygone
mistakes which were denounced at the time as a mathematical opprobrium.
When studying the dependences between random events or variables, Markov, who
possessed a remarkable flair for natural sciences, paid special attention to, and profoundly
investigated an important class of sequences of dependent events which he called chains and
which are now known in science as Markov chains. The English scientist Galton was the first
to consider some particular cases of such kinds of dependences. Already in the last {the 19th}
century, having a rather vast and diverse experimental material at his disposal, he attempted
to convert the Darwinian theories of heredity and evolution into mathematical form. Neither
he, nor the physicists, who also came to construct stochastic patterns similar to Markov
chains almost at the same time as the Russian mathematician did, have offered their precise
definition, or, still less, any satisfactory mathematical analysis of their properties.
The closeness of Markovs ideas to those, having emerged at the same time in diverse
fields of natural sciences, testify to their vitality and to the wide scope of their possible
applications. Indeed, a Markov chain is a stochastic transformation of a usual determinate
process whose dynamic state at a given moment fully defines its further course independently
of all its previous states. More specifically, according to Markov, a chain is such a sequence
of random variables x1, x2, , xn, xn+1 that, after xn takes some definite value, the
probabilities that the subsequent variable xn+1 takes some values are fully determined
independently of the values that the previous variables x1, x2, , xn1 had.
Excluding the limiting case in which the value of the variable xn fully defines the next link,
and the chain is thus transformed into a determinate process, Markov proved that the
dependence between the links weakens with the increase in the distance between them, and
went on to infer that the law of distribution for the variable xn tends to some limiting
stationary law independent from the initial value of x1. In particular, this theorem, that finds
many applications, can, under some very general suppositions, justify the underpinning
principle of the kinetic theory of gases, i.e., that of the uniform distribution of energy among
the phases for established conditions. The extension of the LLN onto Markov chains is a
simple corollary of the fact that the dependence between remote links of the chain is very
weak.
The problem about the limiting applicability of the normal law to the sum of terms
constituting a chain presented more essential difficulties. Applying the method of moments,
Markov also solved it under rather general suppositions. The further generalization of these
results was achieved later, and I shall not dwell on this topic.
Concluding my review of the investigations of Markov himself into dependent variables, I
shall only cite
1) His curious philological phonetic application of chains to studying the alternation of
vowels and consonants in literary works; and
2) One of his last contributions of a somewhat different nature. There, he indicated and
analyzed a stochastic pattern leading to limiting laws of distribution represented by the
Pearsonian curves. He thus revealed the essence of the conditions under which the use of
these curves in statistics was justified.
[6] Only one point is left now, viz., the listing of the main directions of the modern
probability theory that have been developed under the direct influence of the Petersburg
school. First of all, I mention the systematization and detailed study of all the limiting laws
obeyed by the sums of an infinitely increasing number of independent terms, including the
case in which the Liapunov condition does not hold. These investigations threw additional
light on the special importance of the normal law in probability theory, where its role is
similar to that played by the principle of inertia in mechanics. On the other hand, the
Liapunov method of characteristic functions was extended onto the case of several sums so
that the theory of normal correlation became thus justified. Finally, the investigation of
dependent variables that followed along the way paved by Markov proved most fruitful and
important. Here, we now have essentially new methods and results relating to the Markov
chains themselves and to their various generalizations; above all I should cite the so-called
theory of stochastic differential equations which is undoubtedly destined for a great future. It
is not my aim to review modern contributions belonging to this direction; there, the theory of
probability, now remote from its engendering elementary problems on random events in
games of chance, is a powerful mathematical tool for cognizing nature.
For its transformation from a mathematical amusement into a method of natural sciences
the theory of probability is mainly obliged to the Petersburg school which accomplished this
fundamental progress leaving West European mathematicians far behind.
1.{Liapunov left four contributions on the theory of probability, and he actually read a
course in probability for three years, see B.V. Gnedenko, On the work of Liapunov in the
theory of probability. IMI, vol. 12, 1959, pp. 135 160, 1; translated in P.A. Nekrasov, The
Theory of Probability. Berlin, 2004.}
2.{See Note 9 to Bernsteins essay on Chebyshev translated in this book.}
8. Oscar Sheynin. On the Relations between Chebyshev and Markov
. Istoriko-Matematich. Issledovania, to appear
Markov regarded Chebyshev with highest respect and became one of the two editors of the
first (Russian and French) sufficiently complete collection of his works 1. And he sharply
opposed Nekrasov when the latter had begun to belittle the importance of Chebyshevs proof
of the CLT. At the same time, however, the relations between Chebyshev and Markov had
not been serene. I describe the appropriate facts (1), and, in 2, make public the extant part
of Markovs report devoted to the memory of his teacher. Finally, I have to say a few more
words about Nekrasov (3). My paper only deals with the theory of probability and the
MLSq.
1. Chebyshev and Markov
1.1. Chebyshevs memoir (1887). The appearance of Nekrasovs paper (1898a) devoted
to the memory of Chebyshev led to a discord between him and Markov. The latter (1898b)
had accused him of underestimating the Chebyshev memoir and then (1899b) remarked that
Nekrasov, in his paper, had not mentioned it. Here are Nekrasovs pertinent statements
(1898b; 1900, p. 384).
The theorems considered in Chebyshevs memoir are due to Laplace; Chebyshev only
devised a better proof for them.
I set infinitely high store by his immortal memoir [Chebyshev 1867] which is a greatest
contribution to science [to probability theory]. And I consider his memoir [Chebyshev 1887]
as of minor importance since it contains that, which was sufficiently rigorously proved much
earlier and included in generally known treatises 2. It is only interesting as being one of the
successful applications of Chebyshevs great inventions to [ingenuity in considering]
problems exhausted earlier.
Markov did not restrict his activities here to criticizing Nekrasov. I quote the opening lines
of his memoir (1899a):
Chebyshevs memoir (1887) is important. Regrettably, this fact is obscured by two
circumstances, viz., by involved derivations and insufficient rigor of reasoning
And Markov went on to eliminate Chebyshevs shortcomings in the proof of the CLT. In a
subsequent work Markov (1913, p. 321) called Chebyshevs memoir remarkable.
1.2. Bienaym. Two issues connect Bienaym and Chebyshev (Heyde & Seneta 1977;
Gnedenko & Sheynin 1978, pp. 258 and 261 262); Sheynin (1989, pp. 346 348). First, in
1853 Bienaym proved that, in modern notation,
P( x Ex
var x /
where x was the arithmetic mean of a series of observations and > 0. Bienaym had not,
however, isolated his formula from the general context of his memoir.
Markov (1924, p. 92; also in the edition of 1913) called (1) the Bienaym Chebyshev
inequality and mistakenly stated that Bienaym had only indicated the main idea of its proof,
restricted, moreover, by some conditions. Recall, however, that Chebyshev had not only
independently proved that inequality but applied it for deriving most important corollaries to
his LLN.
Second, again in 1853, Bienaym investigated the order of the even moments of sums of
independent observational errors whereas Chebyshev applied the moments for proving the
Markov (1912, p. 218) declared that he had developed and extended the Bienaym
method. Soon afterwards, in the Introduction to the German edition of his treatise [see Ref.
Markov (1924)], he mentioned the remarkable Bienaym Chebyshev method 3. Nekrasov
(1911, p. 436), however, thought that Chebyshev had exhausted the idea of Bienaym and
referred to Chebyshev (1874). It was in this connection that Markov returned to that issue in
1912. Moreover, he justly added that Chebyshev, who had highly respected Bienaym
(Gnedenko & Sheynin 1978, pp. 261 262), had never said anything of the sort.
The method of moments is less effective than the method of characteristic functions and
had not been generally applied in the 20th century (Zolotarev 1999). Already Liapunov (1900,
p. 125 of the Russian translation) had called Markovs proof of the CLT too involved and
unwieldy and Bernstein (1945, p. 427) stated that the method of moments did not
facilitate the proof of the CLT but rather transferred all the difficulties elsewhere.
Nevertheless, that method is still applied in mathematical statistics for solving concrete
problems (Soloviev 1997, p. 12).
1.3. The Method of Least Squares. Markov (1899a, pp. 249 250) criticized Maievsky
(1881) (and other unnamed Russian authors) for an unsuccessful attempt to justify the
optimal property of the arithmetic mean (the Gauss postulate). Maievsky had cited the
Chebyshev form of the LLN whereas Markov remarked that his substantiation was hollow. It
is remarkable that Chebyshev communicated Maievskys manuscript to the physical and
mathematical section of the Petersburg Academy of Sciences (Mandryka 1954, p. 81). It is
unclear, however, whether he actually noted the pertinent attempt.
Anyway, Chebyshev had hardly been interested in the treatment of observations: he (1936,
pp. 249 250) wrongly described the estimation of their precision (Sheynin 1994, p. 336) 4
and he did not describe definitely enough the application of the CLT to the theory of errors.
Before formulating that theorem, he remarked, that, given a large number of observations, it
led to the MLSq; actually, to the normal distribution whose existence is not, however,
necessary for justifying the method. In addition, Gauss had on principle only considered the
treatment of a finite number of observations (and Markov, see above, could have well
indicated this fact also).
1.4. The Offence. And so, while fully understanding the importance of Chebyshevs work,
Markov nevertheless felt it necessary to criticize him directly (1.1) and perhaps indirectly
(1.3). And he also had grounds for feeling himself offended by his teacher. Bernstein (1945,
pp. 416 417) indicated that the path from the Chebyshev inequalities (Chebyshev 1874) to
the CLT had already then, after 1874, been obvious and noted, on p. 419, that Chebyshev,
who had no rivals, was in no hurry either to prove those, only formulated by him inequalities,
or to surmount the last obstacles.
Markov (1884), however, without waiting for Chebyshev, proved these inequalities. Krein
(1951) described their history and I shall only repeat that Stieltjes had independently
substantiated them a bit later, and then (1885) expressed his regret that he did not, at the time,
know about Markovs proof.
After that, Chebyshev, on the contrary, speeded up his stochastic investigations (Bernstein
1945, pp. 419 420 and 423) now collected, together with his other contributions, in vol. 3
of his '
(Complete Works). But he had not even once mentioned
either Markov or Stieltjes. No wonder that Markov voiced his dissatisfaction, although only
in a private letter to the eminent geologist A.P. Karpinsky (1847 1936) (Grodsensky 1987,
pp. 62 63). In 1885 5, as he wrote, Chebyshev published without proof and without
mentioning me his solution, and he is continuing in the same vein.
All this resembles the well-known attitude of Laplace and Gauss to other scientists. Here,
however, is a reasonable opinion about the latter (Biermann 1966, p. 18):
What is forbidden for usual authors, should be really allowed for Gausses, and in any case
we ought to respect the considerations that guided him.
This means that suchlike attitudes are forgiven by future generations only with respect to
Gausses, and therefore to Chebyshev, as I would add.
2. Markovs Report
2.1. Some Explanation. The centenary of Chebyshevs birth occurred in 1921. Liusternik
(1965, p. 23) reported that in the spring of that year Moscow University and the Moscow
Mathematical Society had received an invitation from the Academy of Sciences, then located
in Petrograd ( former, and present-day Petersburg) to participate in a scientific conference
devoted to that occasion. Liusternik, who was present there, indicated that it was convoked
on Markovs initiative.
This statement might be complemented. Here is Markovs own earlier pronouncement
(Grodzensky 1987, p. 63):
This day should have been commemorated by a special grand meeting; however, since
our life is going on in an abnormal form, any celebrations whatsoever are hardly
appropriate.
But he added that he did not exclude such a possibility if only the physical and mathematical
section of the Academy decided that a meeting is necessary.
More than seven reports were read at the conference, Markovs included (Liusternik
(1965, p. 25):
Markov seemed physically weak and old, but his eyes shone provocatively. He spout caustic
hints. He read a report imparting his recollections of Chebyshev. It was regrettably neither
written down, nor taken down short-hand, and all that, what a man, who had for several
decades rubbed shoulders with Chebyshev, was able to say, was irrevocably lost.
A certain text is however partly extant. It was written by Markov and devoted to
Chebyshev, but has neither date nor signature. Markovs authorship is nevertheless doubtless
because the text includes such phrases as in 1883 I was able , and no-one except for
himself could have written them. The text has no reminiscences 6; it is possible, however,
that the organizers of the conference had asked Markov to change somewhat the subject of
his future speech (or that he himself included them later on). In any case, the text was likely
written on the occasion of Chebyshevs centenary.
The extant text is kept at the Archive of the Russian Academy of Sciences, Fond 173,
Inventory 1, sheet 60, and written on a form with the following letterhead:
Deputy Chairman, Vasileostrov district, Petrograd Commercial & Industrial Union.
Vas[ilievsky] Ostr[ov], Nikolaevskaia Embank. 11, Tel. 77627
Markovs references were incomplete, some of them lacking altogether. He abbreviated
many words; his first phrase, for example, ran as follows: Cheb works th prob make up
consid part of his scient activ and are very remark.
2.2. The Text
Chebyshevs works on the theory of probability make up a considerable part of his
scientific activity and are very remarkable. Nevertheless, they had not attracted either foreign
or even Russian scientists. Such an attitude might be explained by the usual understanding of
probability theory as being an applied science for which mathematical rigor is unnecessary.
An additional circumstance is that Chebyshev was concerned not with new propositions of
the theory, but with new methods for establishing those main theorems which are generally
considered as having been quite ascertained by Laplace and Poisson. Although Chebyshev
had been well known in France, and especially in Paris, neither Bertrand (1888), nor
Poincar (1896) mentioned him at all. Czuber, in his essay (1899) on the development of the
theory of probability, twice cited Chebyshevs works, but his indications did not provide any
clear idea about them.
Apart from two short notes, which appeared in the Liouville and Crelle periodicals {in J.
math. pures et appl. and J. reine und angew.Math.}, Chebyshevs first published work is his
[Masters] dissertation Essay on an Elementary Analysis of the Theory of Probability of
1845, which, as stated in its very title, was devoted to probability theory. Note that it
appeared a year earlier than the generally known fundamental work of another of our late
academicians, Buniakovsky (1846). Note also that the teaching of the theory of probability at
Moscow University, where Chebyshev had obtained his higher education, only began in
1850, even later than in Petersburg. In Petersburg {where Chebyshev had been working} the
subject of the dissertation was proposed not by a university professor 7 (who is not {none of
whom are} mentioned either in its Introduction or main text, but by the curator of the
Moscow educational region, Count {S.G.} Stroganov.
{An obvious gap in the text appears here.}
This note had been unnoticed until the beginning of the 1880s when it caused a lively
exchange of opinions among Petersburg scientists, and mostly between me and my respected
teacher, K.A. Posse, whom I am greeting as the eldest student of Chebyshev present at our
meeting 8. Professor Korkin {Chebyshevs student, 1837 1908} even doubted that the
Chebyshev inequality was valid. Finally, in 1883 I was able not only to prove it, but to solve
his problem {Markov 1884a}. And I was just in time because a little later the eminent Dutch
scientist Stieltjes {Krein 1951} arrived at the same inequality and at the problem of moments
not knowing that Chebyshev had published these inequalities in 18739. And Stieltjes proof
fully coincided with mine 10.
My dissertation that appeared in 1884{1884b} prompted Chebyshev to return to his
problem 11 with which a large part of his work accomplished during the last ten years of his
life had been connected. From among nine contributions of that period only two (On the
simple components and Sur les expressions approches de la racine carre) have nothing in
common with the method of moments 12.
the results of Chebyshevs inferences found application in ballistics as witnessed by
Maievskys book (1881) which I have already mentioned and by his reference to the writing
by the French artillery Captain Jouffret. Finally, the German three-volume treatise
Regarding the other ones, it might be assumed that they only constitute a part of a
contemplated series whose implementation was thwarted by his death. Perhaps he intended to
supplement the previous conclusions by estimating the pertinent deviations caused by
finiteness.
Two years separate this period from the previous one: from 10 May 1883 until 8 October
1885 Chebyshev had not communicated a single work to the Academy 13. He concluded by
the memoir Sur les sommes qui dpendent des valeurs positives dune fonction quelconque
communicated in 1894 and published posthumously in 1895. I consider the three first
contributions of that period, closely connected with each other, as being the main ones. These
are (1885), communicated 8 October 1885, the next one, Sur les rsidus intgraux,
communicated 18 November 1886, and, finally, (1887), communicated 10 March 1887.
3. Nekrasov
On Nekrasovs work in the theory of probability, mostly on the CLT, see Seneta (1984),
Soloviev (1997) and Sheynin (2003). Nekrasov apparently proved the CLT for the case of
large deviations, but he attempted to justify his innovation by an arbitrary interpretation of
Chebyshevs formula of the theorem. Chebyshev (1887) had indicated that the integral of the
exponential function of the negative square had any limits whereas Nekrasov (1911, p.
449) declared that Chebyshev had considered variable limits. Liapunovs statement (1901,
pp. 62 63) has been left unnoticed. He explained that he was not interested in the case of
variable limits, but nevertheless adduced some pertinent remarks.
Commentators believe that Nekrasovs unthinkable mistakes and absurd pronouncements
were caused by the burden of his administrative work; by his inadmissible desire to unite
mathematics, religion and philosophy; and even (Mikhailov & Stepanov 1985, p. 225) by a
mental illness. It is not amiss to add, however, that Nekrasov was indeed able to absorb the
ideas of the religious philosopher V.S. Soloviev, who (Radlov 1900, p. 786) had stated, that
philosophy offers its hand to religion; that the basis of veritable knowledge is mystic or
religious perception; and that (p. 787) the system of veritable knowledge is a thorough
synthesis of theology, rational philosophy and positive science. So had not Nekrasov
subordinated mathematics to religion and (poorly understood) philosophy?
It was the same Nekrasov, who had previously been quite another person, and, in 1896,
wrote a marginal note on the text of Chuprovs candidate dissertation (Sheynin 1990, p. 85):
We [mathematicians] are content with notions concerning force, space, time, probability, etc.
Concerning these same subjects philosophers have written full volumes of no use for
physicists or mathematicians. From our point of view, Mill, Kant and others are not better
but worse than Aristotle, Plato, Descartes, Leibniz
1. Markov Jr (Markov 1951, pp. 605 606) described a remarkable episode connected
with the preparation of Chebyshevs collected works (vols 1 2, 1899 and 1907, in Russian
and French; French edition reprinted: New York, 1962). In 1902, after a hardly justified
revocation of the election of Maxim Gorky to honorary membership of the Academy of
Sciences, Markov sent in his resignation from that body. His application was refused and he
resumed the editorial work apparently having been afraid of handing over that responsible
work to someone else. And it was Chebyshev on whose proposal Markov was elected, in
1886, adjunct of the Academy of Sciences.
2. Here, Nekrasov referred to Laurent (1873, pp. 144 165) who nevertheless (on pp. 144
145) had not estimated the error of his approximations. Can it be that Nekrasov deliberately
indicated wrong page numbers?
3. Seneta (1998, p. 297) somehow decided that Markov had defended Bienayms priority
in spite of his volatile nature. Markov, however, always acted in accord with what he
considered (although sometimes mistakenly) just; he admired Chebyshevs talent, but would
have never attributed to his teacher more than the deserved. True, Markov (1914, p. 162)
once remarked that he sometimes called the method of moments only after Chebyshev, but
this did not change his attitude towards Bienaym.
4. Chebyshev stated that the Gauss formula for estimating the variance was based either on
the existence of a large number of observations, or on the law of hypotheses that does not
deserve great confidence. By that law he apparently meant the existence of a uniform prior
distribution of observational errors only assumed by Gauss in his first justification of the
MLSq; that assumption was not even required because of the introduced at the same time
postulate of the arithmetic mean. Then, Gauss never considered the case of a large number of
observations, a fact that also bears on the last sentence of my 1.3 and could have been
indicated by Markov, see above. Finally, when comparing the Gauss formula with its
previous version, Chebyshev, unlike Gauss (whom he never mentioned), had not indicated
that it provided an unbiased estimate of the variance.
5. Markov had in mind the memoir (1885) read 8 October 1885. Indeed, he was not
mentioned there.
6. There were no caustic hints either. These could have been levelled, first and foremost,
against Nekrasov.
7. Markov hardly meant Buniakovsky who had become professor in 1846.
8. Had Posse been absent, which was unlikely, Markov would have omitted the
appropriate phrase. Alternatively, he could have written his text after reading the report.
9. The plural form (inequalities) is correct, the singular form (above) is wrong.
10. The next lines up to the three-volume treatise were crossed out. The phrase
Regarding the other ones apparently meant Regarding the other seven memoirs. I
mentioned Maievsky (see below) in 1.3, but he is not cited in the previous lines of the
extant text.
11. Cf. Bernsteins relevant pronouncement in 1.4.
12. A chronological list of Chebyshevs works is included in vol. 5 of his Complete Works
and it only contains the second of those memoirs (published in 1889). On the other hand, it
has another memoir of the same year, Sur le systme articul le plus simple obviously
not connected at all with the method of moments. All nine of Chebyshevs last memoirs were
included in the first edition of Chebyshevs collected works, see Note 1.
13. Cf. Note 5.
Bernstein, S.N. (1945), On Chebyshevs work on the theory of probability.Translated in
this book.
Bertrand, J. (1888), Calcul des probabilits. Paris. Second edition, 1907; reprint, 1970.
Biermann, K.-R. (1966), ber die Beziehungen zwischen Gauss und Bessel. Gauss-Ges.
Gttingen, Bd. 3, pp. 7 20.
Buniakovsky, V.Ya. (1846), (
(Principles of Mathematical Theory of Probability). Psb.
Chebyshev, P.L. (1867), Des valeurs moyennnes. J. math. pures et appl., t. 12, pp. 177
--- (1874), Sur les valeurs limites des intgrales. Ibidem, t. 19, pp. 157 160.
--- (1885, in Russian), Sur la rpresentation des valeurs limites des intgrales par les
rsidus intgraux. Acta Math., t. 9, 1886, pp. 35 56.
--- (1887, in Russian), Sur deux thormes relatifs aux probabilits. Ibidem, t. 14, 1890
--- (1936),
1879 1880 (Theory of Probability. Lectures
of 1879 1880). M. Translation: Berlin, 2004.
(Complete Works), vols 1 5. M. L.
--- (1946 1951), '
Czuber, E. (1899), Die Entwicklung der Wahrscheinlichkeitstheorie und ihrer
Anwendungen. Jahresbericht Deutscher Mathematiker-Vereinigung, Bd. 7, No. 2. The whole
Gnedenko, B.V., Sheynin, O.B. (1978, in Russian), Theory of probability. In Kolmogorov,
A.N., Yushkevich, A.P., Editors (1992), Mathematics of the 19th Century, vol. 1. Basel, pp.
211 288. Reprinted in 2001.
Grodzensky, S.Ya. (1987), Markov. M. (R)
Heyde, C.C., Seneta, E. (1977), Bienaym. New York.
Krein, M.G. (1951), Chebyshevs and Markovs ideas in the theory of the limiting values
of integrals and their further development. Uspekhi Matematich. Nauk, vol. 6, No. 4, pp. 3
Laurent, H. (1873), Trait du calcul des probabilits. Paris.
Liapunov, A.M. (1900), Sur une proposition de la thorie des probabilits. Izvestia Imp.
Akad.Nauk St. Ptersb., sr. 5, t. 13, pp. 359 386.
--- (1901), Answer to Nekrasov. Zapiski Kharkov Univ., vol. 3, pp. 51 63.
Liusternik, L.A. (1965), Report at the jubilee sitting of the Moscow Mathematical Society.
Uspekhi Matematich. Nauk, vol. 20, No. 3 (123), pp. 21 30. (R) That periodical is being
translated as Russian Math. Surveys.
Maievsky, N.V. (1881),
(Explication of the Method of
Least Squares and of Its Applications Mostly for Investigating the Results of Artillery
Firing). Psb.
Mandryka, A.P. (1954), Maievsky. Moscow. (R)
Markov, A.A. [Sr] (1884a), Proof of some of Chebyshevs inequalities. In authors book
$ !
(Sel. Works on the Theory of Continued Fractions and the Theory of
Functions Least Deviating from Zero). M. L., 1948, pp. 15 24.
. (On
--- (1884b), (
Some Applications of Algebraic Continued Fractions). Doctors Dissertation. Psb.
--- (1899a), The law of large numbers and the method of least squares. In Markov (1951,
pp. 233 251). Translated in this book.
(1899b), An answer [to P.A. Nekrasov]. Izvestia Fiziko-Matematich. Obshchestva Imp.
Kazan Univ., ser. 2, vol. 9, No. 3, pp. 41 43.
--- (1912), A rebuke to P.A. Nekrasov. MS, vol. 28, pp. 215 227.
--- (1913), The theorem on the limit of probability for the Liapunov case. In Markov
(1951, pp. 321 338).
--- (1914), Bicentennial of the law of large numbers. In Ondar, Kh. O., Editor (1977, in
Russian). Correspondence between Markov and Chuprov, etc. New York, 1981, pp. 158
(Calculus of Probability). M. Previous editions:
1900, 1908, 1913. German translation of second edition: Leipzig Berlin, 1912.
(1951),
(Sel. Works). No place.
Markov, A.A. [Jr] (1951), Biography of Markov Sr. In Markov (1951, pp. 599 613).
Translated in this book.
Mikhailov, G.K., Stepanov, S.Ya. (1985), On the history of the problem of the rotation of
a solid about a fixed point. IMI, vol. 28, pp. 223 246. (R)
Nekrasov, P.A. (1898a), General properties of mass independent phenomena, etc. MS, vol.
20, pp. 431 442.
--- (1898b), Letter to Markov of 7 Oct. 1898. Archive of Russian Academy of Sciences,
Fond 173, Inventory 1, Delo 53, No. 1.
--- (1900), Concerning Markov (1899b). MS, vol. 21, pp. 379 386.
(1911), On the principles of the law of large numbers, the method of least squares and
statistics. MS, vol. 27, pp. 433 451.
Poincar, H. (1896), Calcul des probabilits. Paris. Second edition, 1912, reprinted in
1923 and 1987.
Radlov, E.L. (1900), V.S. Soloviev. His religious and philosophical views. +
(Brockhaus & Efron Enc. Dict.), halfvol. 60, pp. 785 792.
Seneta, E. (1984), The central limit theorem and linear least squares in pre-revolutionary
Russia. Math. Scientist, vol. 9, pp. 37 77.
--- (1998), Bienaym (1798 1878): Criticality, inequality, and internationalization.
Intern. Stat. Rev., vol. 66, pp. 291 301.
Sheynin, O. (1990, in Russian), Chuprov. Gttingen, 1996.
--- (1989), Markovs work on probability. Arch. Hist. Ex. Sci., vol. 39, pp. 337 377; vol.
40, p. 387.
--- (1994), Chebyshevs lectures on the theory of probability. Ibidem, vol. 46, pp. 321
--- (2003), Nekrasovs work on probability: the background. Arch. Hist. Ex. Sci., vol. 57,
Soloviev, A.D. (1997), Nekrasov and the central limit theorem of the theory of
probability.IMI, ser. 2, vol. 2 (37), pp. 9 22. Transl.: DHS 2799, 2004, pp. 199 208.
Stieltjes, T.J. (1885), Note loccasion de la rclamation de M. Markoff. Oeuvr. Compl., t.
2. Berlin, 1993, pp. 430 431 (518 519). Double paging.
Zolotarev, V.M. (1999), The method of moments. In Prokhorov, Yu.V., Editor (1999),
(Probability and Math.
Statistics. An Encyclopedia). M., p. 360.
Translations of the following Russian sources are available in P.A. Nekrasov, Theory of
Probability. Berlin, 2004: Liapunov (1901); Markov (1898; 1899b; 1912; 1913; Nekrasov
(all those referred to).
9. A.M. Liapunov. On the Gauss Formula for Estimating the Measure of Precision of
Manuscript, first published by the present translator in Istoriko-Matematich. Issledovania,
vol. 20, 1975, pp. 319 328.
Suppose that n observations are made to determine m magnitudes x, y, z, and they
directly provide the values of the following n linear functions
aix + biy + ciz + (i = 1, 2, , n).
Denote these values given by observations by g1, g2, , gn. We will have
aix + biy + ciz + = gi+
(i = 1, 2, , n)
where 1, 2, , n are the errors made.
Assuming that among the n linear functions under consideration there are at least m
mutually independent ones (n > m), and that the approximate equalities
aix + biy + ciz + = gi (i = 1, 2, , n)
are free from constant error, i.e., that the expectations of the possible error in each
observation are equal to zero, we apply the MLSq to determine the unknowns x, y, z,
To this end, we denote the expectation of the square of the possible error of the i-th
observation {cf. formula (4) below} by k/pi, and, assuming that the weights of the
observations p1, p2, , pn are known, we look for x, y, z, under the condition that the
pi(aix + biy + ciz + gi)2
is minimum.
Let , , , be the values of the unknowns thus determined. Then magnitudes
ai + bi + ci + gi =
will represent the corrections of the observations obtained by the MLSq.
Suppose that it is required to determine the magnitude k on which the measure of precision
pi / 2k of each observation depends. For this purpose Gauss 2 proposes to use the formula
k=
which, however, is not yet rigorously established. It is not therefore amiss to indicate the
following proposition on which it might be founded: Suppose that i is the possible error of
the i-th observation and i is the corresponding correction obtained by the MLSq. If, no
matter how long the observations be continued (supposing that it is possible to continue them
indefinitely), the ratio
/(E
remains less than some limit, then the probability of the inequality
pi i
k| < ,
no matter how small the positive number
making n sufficiently large.
Denote
piai i = Ea,
pibi i = Eb,
Then, expressing the magnitudes
would be, can be made arbitrarily close to 1 by
pici i = Ec, etc.
by means of
(Ea; Eb; Ec; ),
we shall have, as it is known,
where (Ea; Eb; Ec; ) stands for some positive definite quadratic form of variables Ea, Eb,
Ec; 3
Hence we find that
k| < |
k | + /n.
Consequently,
k| < E|
k | + E /n.
However, it is known that
E = mk.4
On the other hand, noting that
= k/pi
and supposing that E
k | < [E(
= k2qi/pi2, we have 5
k )2]1/2
k | < (k//n) (
qi / n ) 1 .
We thus obtain the inequality
k| < (k//n) (
qi / n ) 1 + mk/n
from which, making use of the Chebyshev reasoning 6, our proposition is indeed deduced.
Suppose that
) = qi < L.
The obtained inequality provides
k| < (k//n) L 1 + mk/n.
Evidently, however,
k| > (1 P)
where P is the probability of the inequality
k| .
Therefore, we have
(1 P) < (k//n) L 1 + mk/n
P > 1 [k/( /n)] [ L 1 + (m//n)]
which indeed proves the theorem.
Basing our reasoning on this theorem, we may consider each expression of the type
f (n)
as an approximate expression for k. Here, f(n) is any function of n for which
lim [f(n)/n] = 1 as n = .
However, from among all such expressions there is only one that will be free from a constant
error; namely that, for which
= k.
This expression is f(n) = n m which gives us
Notes and Commentary by Translator
1. According to Gauss [2, 178 179], the measure of precision, h, is the parameter of
the normal distribution
1(x)
= (h/
) exp ( h2x2).
In 1823 he did not retain this term and the normal law itself lost its uniqueness, and,
furthermore, appeared, in 7 and 9, in another form:
= [1/(h
)] exp ( x2/h2).
On the other hand, Gauss formally introduced weight only in [3, 7].
2. See [3, 38 and 39].
3. In notation of the classical error theory formula (2) becomes
pi i2 =
p i2 {[paa](x )2 + [pbb](y )2 + [pac](z )2 +
2[pab] (x ) (y ) + 2[pac] (x ) (z ) + 2[pbc] (y )(z )}
where, for example, [pab] is the sum of terms piaibi and, without loss of generality, the
number of the unknowns is three. Magnitudes [paa], [pab], , [pcc] are the coefficients of
the appropriate system of normal equations.
4. My formula (i) leads to [6, pp. 81 89]
E(
pi i2
pi i2) = k{Q11[paa] + Q12[pab] + Q13[pac] +
Q21[pab] + Q22[pbb] + Q23[pbc] + Q31[pac] + Q32[pbb] + Q33[pcc]}.
By definition of the magnitudes Qi j the right side is equal to 3k (or, generally, to mk), hence
Liapunovs formula (3).
5. The next line does not directly follow from the context, but it represents a particular
case of Liapunovs inequalities [7, p. 112].
6. Liapunov apparently thought that Chebyshevs reasoning was his (or, rather, the
Bienaym Chebyshev) inequality. Formula (6) below can be deduced at once by means of
elementary transformations made while proving it [5, 32]. Let P( ) = p. Then
1 p = P( > ) (1/ )
|x| (x) dx (1/ )E| |
|x |>
where (x) is the density law of the random variable .
Seneta [8, p. 54] in essence noted that, more to the point, (ii) was the Markov inequality
for positive random variables.
Liapunov possibly wrote his note at about the time when he was working on his main
papers on probability theory; that is, somewhat before 1900, see Gnedenko [4]. Suppose that
n observations g1, g2, , gn are made to determine m unknowns, x, y, In the ideal case,
aix + biy + + gi = 0, i = 1, 2, , n.
Residual free terms i appearing after solving these equations by least squares can be used to
estimate the precision of observations. In Liapunovs notation, see formula (1), the estimator
of the squared mean square error of unit weight is [3, 38]
Liapunov mistakenly thought that this formula was not yet established rigorously and set
out to prove it anew, and to determine whether it was good enough as an estimator. He cited
Gauss but had not mentioned any definite contribution so that possibly he did not then read
the Theoria combinationis [3]. It is likely, however, that he acquainted himself with this
memoir later (for example, on Markovs advice) and that, accordingly, he abstained from
publishing his manuscript.
Liapunov undoubtedly came to know the Gauss formula from his teacher, Chebyshev,
whose lectures on probability [1] were eventually published from notes taken by him,
Liapunov. I [9] studied these lectures and indicated, in 5.3, that Chebyshev had not treated
the Gauss formula adequately. Now I add that, consequently, Liapunov could have indeed
inferred that it was not established rigorously. Cf. also Note 1 above.
What had Liapunov achieved? Yes, he proved the Gauss formula anew. And he also
derived inequality (7) from which it followed that, for any law of error, the estimator (8) of
the variance was consistent. This, however, was already evident from Gauss findings [3,
40]. Seneta [8, p. 54] indicated this fact, but he cited Gauss additional formula that only
pertained to the normal distribution.
1. Chebyshev, P.L. (1936),
(Theory of Probability). Lectures read
1879/1880 as written down by A.M. Liapunov. M. L. Translation: Berlin, 2004.
2. Gauss, C.F. (1809), Theoria motus, etc. German translation of extract in Gauss, C.F.
(1887), Abhandlungen zur Methode der kleinsten Quadrate. Hrsg. A. Boersch & P. Simon.
Vaduz, 1998, pp. 92 117.
3. --- (1823), Theoria combinationis, etc. German translation: Ibidem, pp. 1 53.
4. Gnedenko, B.V. (1959), On the work of Liapunov in the theory of probability. IMI, vol.
12, 1959, pp. 135 160. Translated in Nekrasov, P.A. (2004), Theory of Probability. Berlin.
5. --- (1968), Lehrbuch der Wahrscheinlichkeitsrechnung. Berlin. Several previous
Russian editions.
6. Helmert, F.R. (1872), Die Ausgleichungsrechnung, etc. Leipzig. Later editions: 1907,
7. Heyde, C.C., Seneta, E. (1977), I.J. Bienaym. New York.
8. Seneta, E. (1984), The central limit problem and linear least squares in prerevolutionary Russia: the background. Math. Scientist, vol. 9, pp. 37 77.
9. Sheynin, O. (1994), Chebyshevs lectures on the theory of probability. Arch. Hist. Ex.
Sci., vol. 46, pp. 321 340.
10. A.A. Markov. The Law of Large Numbers and the Method of Least Squares
{This memoir consists of extracts from letters written to Professor A.V. Vasiliev}
Chebyshevs memoir [2] is important. Regrettably, its importance is obscured by two
circumstances, viz., by involved derivations and insufficient rigor of reasoning. The theorem,
that Chebyshev proves there, has for a long time been considered true, but it was established
by means of extremely non-rigorous methods. I do not say proved, since I do not admit nonrigorous demonstrations when perceiving no possibility of making them rigorous.
So, Chebyshevs aim was not to establish the theorem, even if non-rigorously, but to prove
it. Its well-known derivation is not rigorous but easy. On the contrary, Chebyshevs
demonstration is very involved since it is based on preliminary investigations. Therefore, its
advantage over the previous proof can only consist in rigor of reasoning. However,
Chebyshevs derivation is expounded in such a way that its rigor may be doubted. A question
therefore arises, whether Chebyshevs proof is distinguished from the previous one not in
essence, but only by needless complexity, or can it be made rigorous. Your essay on the
works of Chebyshev [10] has strengthened my long-standing desire to simplify, and, at the
same time, to make his analysis quite rigorous.
I begin with the Theorem on expectations that makes up the main subject of his memoir.
It can be formulated thus: If the expectations of the independent magnitudes
x1, x2, , xn
are zero, and those of the power xnk for any integral positive number k remain finite as n 0
1, then, under the last-mentioned condition, each of the differences
x + x2 + ... + xn
2 m/2
where m is an integral positive number and
Am = (2
t mexp( t2)dt.
I provide an elementary proof of this proposition 1.According to the well-known
generalization of the Newton formula, we have
x1 + x2 + ... + xn
m!
S 1 , 2 ,..., i
1! 2!... i ! ( n )m
are integral positive numbers (not zeros) satisfying the condition
and S represents a symmetric function of numbers (1) that can be determined by one of its
terms,
x1 1 x2 2 ... xi i .
The number of the terms of this kind is not larger than
n(n 1)(n 2) (n i + 1).
It is equal to this product when all the numbers (3) are different, and less than it otherwise.
Turning now to expectations, we find that the expectation of
is equal to the sum
Here, in accord with one of the conditions of the theorem, all such terms, in which any of
the numbers (3) is equal to unity, vanish. Therefore, only such terms remain for which each
of these numbers is not less than 2. Supposing therefore that
2, ,
we conclude from equality (4) that the maximal value of i in the remaining terms will be (m
1)/2 for odd values of m, and m/2 for its even values. It is also important to note that for an
even m the number i only attains its maximal value m/2 if
Otherwise, i < m/2.
On the other hand, on the strength of one of the conditions of the theorem, the ratio
ES/[n(n 1)(n 2) (n i + 1)]
should remain finite as n
. Therefore, for i < m/2 each of the ratios ES/( n)m should tend
to zero as n
. We infer at once that, as n
, the expectation of the expression
should tend to zero if m is not even. Otherwise, once more, of course, as n
(m!/2m/2)
ES 2, 2,...,2
nm / 2
Let us compare now
and E 1
Denote for the sake of brevity Exi2 = ci, then
c1 + c2 + ... + cn
As to the expectation of S 2, 2,...,2 , it is equal to a symmetric function of the numbers
c1, c2, , cn
determined by one of its terms, c1 c2 cn. Applying now the generalized Newton formula to
, ,...,
( m / 2)! C 1 2 j
1! 2!... j ! n m / 2
where 1, 2, , j are integral positive numbers (not zeros) obeying the condition
1 + 2 + + j = m/2
and C denotes a symmetric function of numbers (5) that can be determined by one of its
terms c1 1 c2 2 ...c j j .
Condition (7) shows that j < m/2 for all the expressions (8) under our consideration
excepting one, C1, 1, , 1, which coincides with ES2, 2, , 2. Therefore, all the terms of the sum
in (6) tend to zero as n
except for one of them,
(m/2)![C1, , 1/nm/2] = (m/2)![ES2, , 2/nm/2].
It follows that the difference
(m/2)! [ES2, , 2/nm/2]
tends to zero as n
. Comparing this result with the one obtained above, we infer that, as n
, the difference (2), where
= 1 3 5 (m 1) = (2m/2/
Am = m / 2
2 ( m / 2)!
t mexp( t 2)dt
should tend to zero. Here, m is any given even positive number.
The theorem is thus proved for even values of m. For its odd values it was proved above
because in this case the last integral vanishes and
lim E 1
= 0 as n
To arrive, after all, to the well-known limiting probability expressed by the
exp( x2) dx
we must transform the theorem of the first letter into the following
Theorem. As n
, each difference
2 E ( x1 + x2 + ... + xn )
t mexp( t2)dt
tends to zero. This transformation does not present any difficulties but
demands a new condition; namely, we must assume that
E(x1 + x2 + + xn)2
increases unboundedly with n and in such a manner that its ratio to n does not tend to zero;
or, better, cannot be arbitrarily small. This condition also plays an important part in the long
ago known derivation of an approximate expression of the probability 3.
I shall show, making use of a particular example, that, if this condition is violated, the
integral (9) does not represent either the approximate value, or even the limit of probability.
Let the values of xk be 1/Nk and 1 with probabilities Nk/(Nk + 1) and 1/(Nk + 1) respectively.
Let also
(1/N1), (1/N2),
be a convergent series of positive numbers whose sum is less than 1. Under these conditions
it is easy to express the probability that
x1 + x2 + + xn < 0.
Indeed, this sum will be negative if all its terms (1) are negative, and
positive if any of its terms is positive, i.e., equal to 1. Therefore, according to
the multiplication theorem, the probability that (12) is fulfilled is expressed by
[N1/(N1 + 1)] [N2/(N2 + 1)] [Nn/(Nn + 1)]
{1 [1/((N1 + 1)]} {1 [1/((N2 + 1)]} {1 [1/((Nn + 1)]} =
1/{[1 + (1/N1)] [1 + (1/N2)] [1 + (1/Nn)]}.
It follows that the probability sought is higher than
1 [1/((N1 + 1)] [1/((N2 + 1)] [1/((Nn + 1)]
but lower than
1/[1 + (1/N1) + (1/N2) + + (1/Nn)].
Until now, the numbers
N1, N2, , Nn
were restricted by one condition,
1 (1/N1) + (1/N2) +
to which we may add the following one:
1/2 [1/((N1 + 1)] + [1/((N2 + 1)] +
Thus, as n
, the probability of condition (12) being fulfilled approaches a
limit larger than 1/2. On the contrary, this limit will be less than 1/2 once the
numbers (14) are chosen in such a manner that
1 = (1/N1) + (1/N2) +
For example, if
Nk = (k + 1)(k + 2) 1,
[1/(N1 + 1)] + [1/(N2 + 1)] + = [1/(2 3)] + [1/(3 4)] + [1/(4 5)] + = 1/2
and the probability considered has as its limit a number larger than 1/2; if,
however, Nk = k(k + 1), the limit of the probability is less than 1/2.
This result shows that the Chebyshev theorem is not applicable to our
example: according to the theorem, the limit of the probability for the
condition (12) to be fulfilled must be equal to unity.
Turning now to the expectation (10) we see that it is equal to
(1/N1) + (1/N2) + + (1/Nn)
. This indeed is the reason why the Chebyshev
and remains finite as n
theorem is not applicable here.
To form a better judgement on the situation, it seems useful to me to dwell in addition on a
modification of my example. I have previously supposed that the positive and the negative
values of each number xk are neither equal in absolute value nor equally probable. However,
it is usually assumed that observational errors only differing in their signs are equally
probable, and I modify my example accordingly.
Suppose that the values of xk are
1, ( 1/Nk), (1/Nk), 1
with probabilities
1/[2(Nk+ 1)], Nk/[2(Nk+ 1)], Nk/[2(Nk+ 1)], 1/[2(Nk+ 1)]
respectively. Then the expectation of xn2 will still be
[1/(Nk+ 1)][1 + (1/(Nk)] = 1/Nk.
Denote the sum of the series (11), which we suppose is convergent, by 2t2 and consider the
probability that the sum in (12) is contained within the boundaries
t{2[(1/N1) + (1/N2) + + (1/Nn)]}1/2.
It is easy to show that, once we appropriately choose the series of numbers N1, N2, , the
will not be the limit of this probability.
Indeed, this probability is higher than that for the sum in (12) to be contained within the
[(1/N1) + (1/N2) + + (1/Nn)].
This latter probability is, however, higher than that for all numbers (1) to be either 1 or 1;
i.e., it is larger than the product (13). After establishing this fact, choose
N1 = 2 3, N2 = 3 4, N3 = 4 5, etc.
The sum (15) will then be equal to 1/2; therefore, t = 1/2 and
exp( x ) dx = (2/
exp( x2) dx < 0.53.
The product (13), that represents a magnitude smaller than the probability considered, is
larger than [1 (1/N2) (1/N3) ], and therefore larger than
6[1 (1/3)]/7 = 0.57
It is thus obvious that the integral (9a) does not represent an approximate value of the
probability and cannot be its limit as n
Now, in accord with the wish formulated in your letter, I turn to the MLSq. Without
assuming any definite law of probability for the errors of the separate observations, it is
possible to arrive at the method by issuing from the following propositions:
1) We consider only such approximate equalities, which, according to our assumptions, do
not contain any constant error.
2) To each approximate equality we assign a certain weight and we suppose that the
weights of the different equalities are inversely proportional to the expectations of the
squares of the errors.
3) We estimate the merit of each approximate equality by its weight, and, accordingly, for
each of the unknowns we determine such an equality whose weight is maximal.
To my mind, only this justification of the MLSq is rational; it was indicated by Gauss. I
consider it rational mainly because it does not obscure the conjectural nature of the method.
Keeping to this substantiation, we do not ascribe the ability of providing the most probable,
or the most plausible results to the MLSq and only consider it as a general procedure
furnishing approximate values of the unknowns along with a hypothetical estimation of the
results obtained 6.
Astronomers like another substantiation, according to which the MLSq is presented as
providing the most probable values of the unknowns. Since this {earlier} justification is also
due to Gauss, I feel that it is useful to quote his own opinion formulated in his letter to Bessel
as provided by Czuber [3, p. 289] 7.
It takes one to be very obstinate to persist in considering the most probable hypotheses,
Gauss words notwithstanding. I return to that justification which I believe to be the only
rational one. According to my opinion, it provides everything needed for practice, but it does
not furnish the probable error 8. I believe that this doubtful magnitude is not required; if,
however, it should be determined without fail, the well-known expression for the probability,
exp ( x2)dx, should be assumed for each error separately, independently of whether it was
derived from one or from several observations.
Here, we do not anymore need to determine the limit of the probability, or prove some
inequalities concerning probabilities. However, it is possible to ask whether there exists a
reasonable substantiation for admitting this formula for each observation separately. One of
the relevant grounds known to me is that this admission allegedly tallies with practice, which
is, however, difficult to ascertain. Another one consists in that the error of each observation
might be supposedly considered as a totality of many errors independent of one another.
Then, only a reference to the theorem on the limit of probability is needed to arrive at the
desired formula; such a reasoning is to be found in Poincar [8, p. 181]. However, the
theorem on the limit of probability can only be ascertained with many restrictions; and, in
addition, the idea that an error is a sum of many independent errors should be attributed, in
my opinion, to the realms of fancy.
Let us now pass on to the derivation of the MLSq based on the theorem according to
which the limit of probability is expressed by the well-known integral. Considering the wellknown formula as a limiting expression, and not admitting it for a single observation, we
have no right of admitting it either for 2, or 10, or 100, or, in general, for any given number
of observations. Actually, however, we always treat a given number of observations, and we
may only speak about the limit after artificially inventing their continuation. We are engaged
not in a derivation for the limiting case, to which we come after continuing our observations
until infinity, but by issuing from a finite number of observations. The theorem, according to
which the limit of probability is expressed by a well-known integral, gives no indication of
how far does the probability, in each given case, deviate from this integral. The question
concerning the magnitude of these deviations remains open; they can be quite considerable,
and, what is the main point, different in different cases. Therefore, when considering the
well-known integral as the approximate value of the probability, and making it maximal, we
are nevertheless unable to maintain that the probability itself will at the same time attain its
maximal value. If, however, the value of the integral remains invariable, we cannot assert
that the probability itself will also be such. The theorem on the limit of probability cannot
therefore serve as a proof that the MLSq provides the most plausible results, i.e., such, that
maximal probability is attained for given limits of error, or narrowest limits of error are
attained for a given probability.
Attempts at substantiating the MLSq by issuing from the theorem proved by Chebyshev
[1] are also found in the Russian literature on probability theory. One such endeavour, very
vague in its essence, might be perceived in Maievskys well-known writing [4] 9. There, in
31 and 32, some kind of a principle of arithmetic mean is considered. In 31 the author
busies himself with deriving it for the case of a very large number of observations without
establishing its meaning. And only in 32 is it possible to conjecture that it is actually the
well-known Gauss assumption that the arithmetic mean of the observed magnitude represents
the most probable value of the measured unknown 10. However, while considering the
reasoning in 31, we ascertain that nothing of the sort follows from it since the author
invariably discusses the arithmetic mean without comparing it with other means.
The conclusion in 31 is that, as the number of observations increases to , the probability
that the arithmetic mean of the observed magnitudes differs from the true value of the
unknown less than by some given quantity, tends to unity. However, nothing follows from
this proposition, since, as the number of observations increases to , the same property also
holds for other (linear) means. Finally, I know that some Russian mathematicians strove to
derive the MLSq by means of a wrong application of the Chebyshev theorems which state
that the probability is larger than a certain magnitude. Bearing in mind this proposition, they
demanded that the probability should exceed a number assigned beforehand, and sought the
narrowest boundaries of the error corresponding to a given inequality for the probability.
They forgot that an uncountable set of different numbers obeys the same inequality and that
the probability can exceed some given number even if the Chebyshev inequality does not
reveal that fact.
Right now I am unable to recall any other justification of the MLSq. It seems that I had a
look at everything, and I can only repeat, that, according to my opinion, the only rational
substantiation is based on a direct study of the expectation of the square of the error. It is to
this formulation that I adhere in my lectures.
Notes, partly by Yu.V. Linnik [6, pp. 654 655]
1. I discovered it long ago, but have not yet published it. M. [It advantageously differs
from the not altogether correct one due to Chebyshev. L.]
2. A similar reasoning is in [8, pp. 168 186]. M.
3. Markov offered such a derivation in [5]. L.
4. An interesting but more complicated example can be found in the well-known book [9].
It is also possible to compile such examples in which the expectations of the squares
represent divergent series. M.
5. The examples provided by Markov represent cases in which a condition, now called
maximal disregard, is violated. At the end of his paper, Markov puts forward illustrations
where it is not violated, but the expectations of the squares constitute divergent series. These
examples were needed for the second part of the paper (for the letter dated 5 October 1898),
where the author criticizes the substantiation of the normal law of observational errors by
means of the hypothesis of elementary errors. L.
6. In simplest cases, the Markov principle is applied for the derivation of the well-known
formulas of the MLSq in the following way. Suppose that linear equations
+ ai2
+ + ar1 r, i = 1, 2, , n
are measured so as to determine the unknown magnitudes 1, 2, , r.
Denoting the errors of observation by i, we obtain random variables yi,
yi =
+ + ar1 r.
Let now E i = 0 and var i = 2/pi where pi are the weights of the observations. Then we
shall choose the approximate value of k as
1ky1
nkyn
= Tk
under the conditions
1) ETk = k (absence of constant errors) and
2) varTk = min.
7. {See English translation in [7, 1977, p. 287]. In this letter dated 1839, Gauss maintained
that the maximal probability of an estimate of an unknown was less important than its least
variance.}
8. The calculation of the probable error demands that the variance of the observations,
rather than only their relative weights, be known. L.
9. I only mention it for the sake of definiteness. A similar reasoning can be found in other
books as well. M.
10. A different sense is attached to this proposition in modern mathematical statistics (the
method of maximum likelihood). It is assumed here that the observational errors are normally
distributed and independent. Under these conditions it is possible to establish a certain link
between the method indicated and the Markov principle. L.
1. Chebyshev, P.L. (1867), Des valeurs moyennes. J. math. pures et appl., t. 12, pp. 177
184. Russian version appeared at the same time.
2. --- (1887, in Russian), Sur deux thormes relatifs aux probabilits. Acta Math., t. 14,
3. Czuber, E. (1891), Theorie der Beobachtungsfehler. Leipzig.
4. Maievsky, N. (1881),
(Exposition of the
Method of Least Squares, etc.). Psb.
5. Markov, A.A. (1898), Sur les racines de lquation Izvestia AN Psb, sr. 5, t. 9, No.
5, pp. 435 446.
6. --- (1951),
7. Plackett, R.L. (1972), Discovery of the method of least squares. Biometrika, vol. 59, pp.
239 251. Reprinted in Studies Hist. Stat. Prob., vol. 2. Editors, Sir Maurice Kendall & R.L.
Plackett. London, 1977, pp. 279 291.
8. Poincar, H. (1896), Calcul des probabilits. Paris.
9. Poisson, S.-D. (1837), Rechereches sur la probabilit des jugements, etc. Paris.
10. Wassilief, A. (1900), P.L. Tschebyschef und seine wissenschaftlichen Leistungen. In
collection of two articles P.L. Tschebyschef (1900). Leipzig, pp. 3 55. French version in
Boll. Bibl. e Storia Sci. Mat., t. 1, 1898.
11. A.A. Markov. The Extension of the Law of Large Numbers
onto Quantities Depending on Each Other
On the strength of the LLN it is possible to maintain, with probability however close to
certainty, that the arithmetic mean of sufficiently numerous magnitudes will differ arbitrarily
little from the arithmetic mean of their expectations. This law was derived by Chebyshev [1]
who considered the expectation of the square of the difference between the sums of these
magnitudes {xi} and of their expectations. Namely, his deliberations made it clear that the
indicated law must persist in all cases in which the square of that difference {call it Ez2}, as
the number of the magnitudes {n} increases unboundedly, increases slower than the square
of that number, so that the ratio of these squares has zero as its limit {[Ez2/n2] 0}.
Chebyshev restricted his derivations with the simplest and therefore the most important
case of independent magnitudes in which, as he showed, the expectation of the indicated
square {Ez2}, as the number of magnitudes {n} increased unboundedly, can only increase as
rapidly as this number if the expectations of the squares of these magnitudes themselves
{Exi2} remain finite rather than increase unboundedly.
To be sure, even if we restrict our attention to independent magnitudes, Chebyshevs
conditions do not at all exhaust all the cases to which the abovementioned law may be
applied. However, we do not aim at discovering the conditions necessary and sufficient for
the applicability of the LLN; we only indicate that Chebyshevs derivations might also be
extended onto some rather general cases when the magnitudes depend one on another.
1. Above all, we may consider the case in which the connection between the magnitudes is
such that the increase in any one of them leads to a decrease in the expectations of the other
ones. Let us dwell on this case. Suppose that we consider magnitudes
x1, x2, , xn,
with expectations
a1, a2, , an,
Denote for the sake of brevity xi ai = zi. When considering the expectation of the square
(z1 + z2 + + + zn)2 =
z12 + z22 + + zn2 + 2z1z2+ 2z1z3 + + 2zn1zn
we shall apply the well-known proposition that the expectation of a sum is equal to the sum
of the expectations of its terms.
For independent magnitudes the expectations of products such as zl zk are all equal to zero
and the expectation of the square (1.2) is reduced to the sum of the expectations of the
squares of z1, z2, , zn. However, in our case the expectation of each product zl zk is a
negative number so that
E(z1 + z2 + + + zn)2 < Ez12 + Ez22 + + Ezn2.
To convince ourselves in this fact we assume that the totality of numbers
zl , zl , , zl(
arranged in an increasing order represent all the possible values of zl with probabilities
pl , pl , , pl(
respectively. Finally, suppose that, if
zl = zl , zl , , zl( ), Ezk = ak , ak, , ak( ).
In this notation
Ezl zk = pl zl ak + pl2 zl2 ak2 + + pl( ) zl( ) ak(
and the expectations of the magnitudes zl and zk themselves, equal to zero, can be represented
as sums
pl zl + pl2 zl2 + + pl( ) zl( ) and pl ak + pl2 ak2 + + pl( ) ak( ).
And since in our case, in accord with the assumption made,
a3k > a2k > > a( )k,
we have, on the strength of the well-known Chebyshev inequality [3; 5],
pl(i) zl(i) ak(i) <
pl(i) zl(i)
pl(i) ak(i) = 0.
It follows that the LLN is applicable to our case if only the expectation of zn2 remains finite
rather than increases unboundedly with n.
The same inequality (1.3), and, consequently, the same inference about the applicability of
the LLN, can also be easily made in the case in which the expectation of xk for any k
decreases with the increase in the sum
x1+ x2+ + xk1.
To show this, it is sufficient to apply the identity
(z1 + z2 + + + zn)2 = z12 + z22 + + zn2 +
2z1z2 + 2(z1 + z2 )z3 + 2(z1 + z2 + z3)z4 +
The indicated conditions are fulfilled, for example, in the case in which the sum
x1+ x2+ + xn
is equal to the number of white balls among n balls extracted from a container under the
following conditions:
1) The container initially had 2a white balls and 2b balls of other colors;
2) The drawn balls are not returned;
3) When (a + b) balls are left in the container, a white balls and b balls of other colors are
added to them.
In this case, just as in the well-known instance in which the ratio of the number of white
balls to all the balls in the container, a/(a + b), remains invariable, we may state, on the
strength of the LLN, with probability however close to certainty, that the ratio of the number
of the appeared white balls to the number of all the drawn balls will differ from a/(a + b) less
than by any given magnitude.
2. Repeating that we only provide sufficient but not necessary conditions, we shall dwell
now on one such case onto which Chebyshevs conclusions might be extended, on the case
in which the influence of the magnitudes (1.1) on each other rapidly decreases with the
increase in the distance between them. In our case the sum (1.4) will represent the number of
the occurrences of some event A in n consecutive trials connected in such a manner that at
each trial the probability of A has a quite definite value p if A had occurred at the
immediately previous trial, and another definite value p otherwise, whichever were the
results of the rest previous trials; if, however, the results of all the trials remain indefinite, the
probability of A is the same and equal to p, for each one of them.
Proceeding to consider this case, we denote for the sake of brevity
1 p = q, 1 p = q , 1 p = q
and we remark that the numbers p, p and p are connected one with another by a simple
p = pp + qp
so that any two of these numbers being given, it is not difficult to calculate the third one;
namely,
p = p /(1 p + p ), p = 1 + p p /p, p = p (1 p )/(1 p).
Since xl and xk equal to 0 or 1 are the numbers of the occurrences of the event A at trials l
and k, it is easy to establish the equalities
al = ak = p.
It is not difficult either to convince ourselves that the expectation of
zl zk = xl xk al xk ak xl + ak al
is reduced to Exl xk p2.
As to Exl xk, it is equal to the probability of the occurrence of the event A at both trials l and
k, which, on the strength of the multiplication theorem, can in turn be represented by the
product of p, of the probability of the occurrence of the event at trial l, by some number
Rkl, by the probability of the occurrence of A at trial k when it is known that it occurred at
trial l.
Having thus arrived at the equality
Ezl zk = p (Rkl p),
we ought to derive Rkl, which only depends, as it is not difficult to convince ourselves, on the
difference (k l) and can therefore be more simply denoted as Rkl.
If k l = 1, then, owing to our conditions, we have
R1 = p .
Then, it is not difficult to calculate consecutively
R2 = R1 p + (1 R1)p = p p + q p , R3 = R2 p + (1 p2)p , ,
Rm+1 = Rm p + (1 Rm)p = p + (p p )Rm.
Rm+1 = p + (p p )Rm
belongs to those, for whose solution it is not difficult to indicate a general formula; namely,
it leads to
Rm = p + C(p p )m
where the constant C is determined by the condition (2.1) and occurs to be equal to 1 p = q.
And so,
Ezl zk = pq (p p )kl
and, consequently,
E[(z1 + z2 + + + zk1) zk] = pq [(p p ) + (p p )2 + + (p p )k1].
It follows that, if p < p , the left side of this equality is negative, so that
E(z1 + z2 + + + zn)2 < npq.
If, however, p > p , then
E[(z1 + z2 + + + zk1) zk] <
and therefore
pq ( p p)
1 p + p
E(z1 + z2 + + + zn)2 < npq 1 +
2( p p)
1 + p p
< npq
We thus obtained inequalities which clearly show that the LLN is applicable to this case.
We may therefore maintain, with probability however close to certainty, that, given a
sufficiently large number of our trials, the ratio of the number of the occurrences of the event
A to the number of the trials will differ from p less than by any given number.
3. The expression for the expectation of the product zl zk found in 2 can be derived when
issuing from general formulas with which we shall deal now and which can serve as a
foundation for further investigations. Keeping to the assumptions of 2, denote the
probability that the event A occurs exactly m times in the first k trials by Pm,k. We may
Pm,k = Vm,k + Um,k
where both magnitudes on the right side are the same as Pm,k subject, however, to the
1) Um,k is the same as Pm,k if A did not occur at the last trial;
2) Otherwise, Vm,k is the same as Pm,k.
We shall now introduce three more functions of an arbitrary number connected with
magnitudes Pm,k, Vm,k, and Um,k:
= Uo,k + U1,k + U2,k 2 + + Uk1,k k1,
+ + Vk,k k,
k = V1,k + V2,k
+ + Pk,k k = k +
k = Po,k + P1,k + P2,k
(3.1a)
(3.1b)
(3.1c)
And we shall show now that the function k can be defined as the coefficient of t k in the
development of a rather simple expression in increasing powers of an arbitrary number t. To
this aim, applying the addition and multiplication theorems, we establish two equalities
Um,k = q Vm,k1 + q Um,k1, Vm,k = p Vm1,k1 + p Um1,k1.
Making use of these equalities and returning to (3.1a) and (3.1b), we obtain two equalities
=q
+q
k1,
+p
and, finally, after eliminating one of the functions,
k+1
(p
+q )
+ (p p )
or , we arrive at
k1 =
But, owing to (3.1c), the following equation must also hold:
can be determined as the coefficient of t k in the development of the
A + Bt
1 ( p + q)t + ( p p)t 2
where A and B do not depend on t, in increasing powers of that magnitude. For determining
A and B it remains to consider k at two values of k; let k = 1 and 2, then we easily derive
=q+p ,
= qq + (qp + pq ) + pp
so that o = 1.
And, after developing our fraction in increasing powers of t and restricting the
development to the two first terms, we find that
A + (B + p + q )t =
= 1 + (q + p )t
so that A = 1, B = (p p ) + q q .
It is not difficult to express the second of these equalities as
B = (p p ) (q + p).
And so, finally,
1 + ( p p)( q + p )t
This formula can serve as a starting point for further investigations, see my paper [6]. In
particular, it is not difficult to apply it for deriving the results of the previous section, but I
shall not dwell on this point.
4. To throw light on our issue, I adduce an example in which the LLN is not applicable.
Suppose that, as in 1, the sum
x1 + x2 + + xn
is equal to the number of white balls from among n balls drawn from a container one after
another. Let, however, the conditions for the drawings be different; we shall fix them as
1) The initial number of white balls in the container was a; and of those of other colors, b.
2) Each drawn ball is returned back together with another one of the same color.
In this case, the increase in the sum
x1 + x2 + + xk1
leads without fail to the increase in the expectation of xk, so that the expectation of
(z1 + z2 + + zk1)zk > 0.
Having in mind the calculation of the expectation of
(z1 + z2 + + zn)2,
we note that it can be expressed by the sum
{m [na/(a + b)]}2Pm,n,a,b
m=0
where P is the probability that m white balls will occur among n of the drawn ones. It is not
difficult to convince ourselves that
Pm,n,a,b = Cn
C a + m 1 Cb + n m 1
C a + b + n 1
The following simple equalities result:
mPm,n,a,b = [na/(a+b)]Pm1,n1,a+1,b,
m(m 1)Pm,n,a,b =
n ( n 1) a ( a + 1)
Pm2,n2,a+2,b.
( a + b) ( a + b + 1)
They lead to
mPm,n,a,b = [na/(a+b)],
Together with an obvious equality
Pm,n,a,b = 1
they make it possible to determine the sought sum (4.1). To prove this we decompose that
sum into three parts,
m(m 1)Pm,n,a,b {[2na/(a + b)] 1}
mPm,n,a,b +
n2[a/(a + b)]2
Pm,n,a,b.
It remains to make simple calculations after which we obtain
{m [na/(a + b)]}2Pm,n,a,b =
nab ( n + a + b) 1
( a + b) 2 ( a + b + 1)
Issuing from this result, we can show that the LLN is not here applicable, so that, given a
sufficiently small , the probability of the inequalities
(m/n) [a/(a + b)]
cannot be arbitrarily close to 1 no matter how large is n.
Indeed, let us denote the probability that these inequalities do not hold by
account that
and take into
{(m/n) [a/(a + b)]}2
cannot be larger than , the larger of the numbers [a/(a + b)]2 and
[b/(a + b)]2. Issuing from the derived formula it is not difficult to ascertain the inequality
which shows that, if only < ab /[(a + b) 2 ( a + b + 1)] , cannot be arbitrarily small.
It is interesting to note that in our example the most probable value of m, which we shall
denote by , is determined by simple inequalities
(a 1)n b + 1 (a + b 2), (a 1)n + a 1 (a + b 2).
In addition, the ratio /n, when n unboundedly increases, has as its limit
not a/(a + b), but (a 1)/(a + b 2).
Nevertheless, we naturally cannot expect that, for large values of n, the ratio m/n will be
arbitrarily close to (a 1)/(a + b 2) since the expectation of [(m/n) g]2 takes its minimal
value at g = a/(a + b), which, as is seen from the formula derived by us, tends, as n , to
! 0.
In the simplest case, in which a = b = 1, the LLN obviously cannot be applied; here, all the
suppositions m = 0, 1, 2, , n have the same probability 1/(n + 1).
5. It is possible to generalize considerably the conclusions of 2. Namely, instead of the number of
the occurrences of some event we may consider the sum of magnitudes connected into a chain in such
a manner that, when one of them takes a definite value, the next ones become independent of those
preceding it. Let
x1, x2, , xk, xk+1,
(5.3a), (5.3b), etc
be an infinite series of magnitudes connected in such a way that, for any k, once xk is known, xk+1 does
not depend on x1, x2, , xk1.
Let also the totality of numbers
, , ",
represent all the possible values of any of our magnitudes and the system
p , p , p ", , p , p , p ", , etc
be the probabilities for xk+1 to have a definite value when xk is given. The first subscript indicates the
value of xk, and the second one, the assumed value of xk+1; for example, if xk = , the probability of
xk+1 = " will be p ".
To prevent too complicated notation and reasoning, we assume that these probabilities do not
depend on k. Let finally
,p
, p"(k),
be the probabilities that xk takes the values (5.2) as long as all the magnitudes x1, x2, x3, remain
indefinite.
The numbers (5.3) should certainly be positive; in addition, they should satisfy equalities
+ = 1, p
+ = 1, etc,
but they are not restricted otherwise. As to the numbers (5.4), they should of course satisfy the
+ p"(k) + = 1
and they cannot be assigned independently of all the values of k. On the contrary, issuing from their
values for k = 1, it is possible to calculate all of them consecutively by means of simple formulas
(k+1)
+ , p
+ , etc.
Turning now to expectations, we denote, as previously, Exk = ak as long as all the magnitudes (5.1)
remain indefinite, and let
A (i), A (i), A"(i),
be the expectations of xk+i when xk = , , ", It is now not difficult to establish the
following equalities:
ak = p
+ p"(k) " + , ak+i = p
A (i)+ p"(k) A"(i) + ,
= p A (i1) + p A (i1) + p " A"(i1) + ,
= p A (i1) + p A (i1) + p " A"(i1) + , etc.
Issuing from these equalities, we shall prove that all the magnitudes
ak+i, A (i), A (i), A"(i),
tend to one and the same limit as i increases unboundedly. To attain this aim, we note, first of all,
that, on the strength of the equalities above, the number ak+i is situated between the largest and the
smallest numbers from among (5.5) all of which, in turn, are situated between the largest and the
smallest numbers from among
(i1)
,A
, A"(i1),
Again issuing from the same equalities, we derive the difference
= (p p ) A
+ (p
p )A
The sum
(p p ) + (p
p ) + = 0.
Therefore, separating the positive and the negative numbers of the system
(p p ), (p
p ),
from each other, and replacing the latter numbers by their absolute values, we will get two totalities
of positive numbers constituting equal sums.
It is also important to note that these sums are less than unity because their terms are less than the
numbers in (5.3a) and (5.3b), respectively. Therefore replacing some of the numbers (5.7) by their
maximal number, and the others, by their minimal number, and denoting by (i1) the difference
between the two last-mentioned numbers, we can establish the inequality
|A
A (i)| < h
where h is the sum of all the positive numbers in system (5.8) and is {therefore} less than unity.
Absolutely the same conclusion can be reached concerning the difference of any of the two
numbers taken from the system (5.5). Therefore, considering (i), the difference between the maximal
and the minimal from among the numbers of that system, instead of A (i) A (i), we can obtain the
<H
where H is some constant number situated between 0 and 1. This inequality shows that
zero as i increases unboundedly.
tends to
It follows that, as i , all the magnitudes (5.6) tend to one and the same limit differing from it less
than by (i). And we also have
< CH i
where C and H are constant numbers and 0 < H < 1.
Turning now to the expectation of
[(x1 a1) + (x2 a2) + + (xn an)]2,
we shall decompose this square into the same terms as in 2 and introduce zk =
xk ak. And,
on the strength of the proven, reasoning absolutely in the same way as we did in 2, we easily arrive
at inequalities
E[zk (z1 + z2 + + zk1)] < D (H + H 2 + + H k1),
E[(x1 a1) + (x2 a2) + + (xn an)]2 < Gn
where D and G are constant numbers.
On the other hand, when comparing the expectations of (5.9) and of
(x1 + x2 + + xn na)2,
where a = lim ak+i, i = , we find that their difference is equal to
[(a1 a) + (a2 a) + + (an a)]2
and persists when n increases unboundedly. Consequently, under this condition, the expectation of
{[(x1 + x2 + + xn)/n] a}2
must tend to zero.
It immediately follows that the LLN is applicable to this case: no matter how small are the positive
numbers and , the probability that the inequalities
< [(x1 + x2 + + xn)/n] a <
take place is higher than (1 ) for all sufficiently large values of n.
And so, independence of magnitudes is not a necessary condition for the existence of the LLN.
Notes and Commentary by N.A. Sapogov [6, pp. 660 662]
1.{As first indicated by Chuprov, this equality can be derived from one of Bohlmanns formulas,
see Ondar (1977, pp. 13 15) who actually provided this derivation. Moreover, the translators of this
source noted that Markovs example was a special case of Plyas urn scheme (Eggenberger & Plya
1923). Also see Feller (1957, 5.2) with a reference to Friedman (1949).}
2. The date of publication of the memoir is given as 1906. S.
Markov considers the conditions for the applicability of the LLN to sums
of dependent random variables Xi, and he only discusses sufficient conditions. His results are based
on a simple but important remark that the law is applicable to Sn if
(varSn /n2)
In the beginning of his work Markov applies a Chebyshev inequality, which, in the needed form, is
formulated thus:
zl(1) zl(2) zl( ); ak(1) ak(2) ak( );
pl(i) 0, i = 1, 2, , , pl(1) + pl(2) + + pl( ) = 1;
pl(i) zl(i) ak(i)
Supposing that
Obviously,
ak(1)
pl(i) ak(i).
> 1, we convince ourselves that this inequalityis valid in the following way.
pl(i) ak(i) ak(
so that there exists such a subscript io, 1 io
ak(i)
pl(i) ak(i) 0, i io, ak(i)
, for which
pl(i) ak(i) < 0, i > io.
{pl(i) zl(i)[ak(i)
(i0 )
pl(i) ak(i)]} =
{pl(i)[ak(i)
{the same}
i = i 0 +1
pl(i) ak(i)]} = 0,
hence (1).
One of Markovs examples of dependent magnitudes to which the LLN was applicable was that of
a simple homogeneous chain; a chain was indeed considered for the first time in this work. In 5 he
studied such a chain under the condition that all transition probabilities p were positive. Note that
this strong restriction was not necessary. Markovs reasoning and all of his conclusions would have
persisted if only he had demanded that there existed at least one such subscript, #, that, for any ,
p# > 0.
From among later works devoted to the LLN for dependent magnitudes I mention [1; 11].
1. Bernstein, S.N. (1918), On the law of large numbers.
4. N.p., 1964, pp. 61 65. (R)
(Coll. Works), vol.
2. Chebyshev, P.L. (1867), Des valeurs moyennes. J. math. pures et appl., t. 12, pp. 177 184.
3. --- (1882), On approximate expression of integrals through other integrals having the same
limits. '
(Complete Works), vol. 3. M. L., 1948, pp. 128 131.
4. Eggenberger, G., Plya, G. (1923), ber die Statistik verketteter Vorgnge. Z. angew. Math.
Mech., Bd. 3, pp. 279 289.
5. Korkine, A.N. (1883), Sur un thorme de Tschbycheff. C.r. Acad. Sci. Paris, t. 96, pp. 326
6. Markov, A.A. (1907, in Russian), Recherches sur un cas remarquable dpreuves dpendantes.
8. Ondar, Kh.O., Editor (1977, in Russian), Correspondence between Markov and Chuprov.
Translated by C. & M. Stein. New York, 1981.
9. Sapogov, N.A. (1950), On the law of large numbers for dependent magnitudes. Izvestia Akad.
Nauk SSSR, ser. Math., vol. 14, No. 2, pp. 145 154.
12. A.A. Markov. The Extension of the Limit Theorems
of the Calculus of Probability onto a Sum of Magnitudes Connected into a Chain
I have shown [3] that the well-known LLN, established by Chebyshev for independent
quantities, extends onto many cases of dependent quantities. I think that special attention
among these instances deserve magnitudes connected into a chain in such a manner, that,
when the value of one of it gets known, subsequent magnitudes become independent from
those preceding it.
When considering one such case, a simplest case as it might be said, I [4] proved for it a
theorem on the limit of expectation from which followed the limit expression of the
probability in the form of the well-known Laplace integral. This result suggests that the
theorem must also take place for other cases of quantities connected into a chain.
My proof was based on a peculiarity of the given instance, viz., on the symmetry of the
expressions sought with respect to p and q; small changes are, however, sufficient for making
the demonstration independent from the indicated feature so that the possibility of extending
such deliberations and the appropriate conclusions onto other cases becomes clear. Without
returning to the studied instance, we shall consider another, more general case and thus
ascertain the proof of a general nature.
be a series of quantities connected into a chain. Suppose also that only three values,
are possible for each of them, and that the first, the second, and the third triplet of the system
of numbers
p, q, r; p1, q1, r1; p2, q2, r2
represent the probabilities of xk+1 = 1, 0, and 1 when xk = 1, 0, and 1 respectively. Of
course, these probabilities cannot be negative and must satisfy the conditions
p + q+ r = 1, p1 + q1 + r1 = 1, p2 + q2+ r2 = 1.
We shall suppose, in addition, that none of them is equal to unity; it is especially important
to assume this with respect to p, q1, and r2 so that our series (1.1) will not be a simple
repetition of the same number. Finally, we must introduce three more numbers,
P, Q, R,
the probabilities of x1 taking values (1.2) respectively.
Under these conditions we shall consider the probabilities of various assumptions about
the magnitude of the sum
for any given n, beginning with small values of this parameter, then going over to its large
values and to n
. If n = 1, our sum is reduced to one number, x1, and, accordingly, it can
only take three values (1.2) with probabilities (1.4).
For n = 2 we have a sum of two numbers, x1 + x2, for which, as is not difficult to see, five
Pp, Pq + Qp1, Pr + Qq1 + Rp2, Qr1 + Rr2, Rr2,
respectively are possible. For the sum (x1 + x2 + x3) there are already seven values [] with
probabilities [] respectively.
Increasing now the number n consecutively by unity, and acting as previously in the article
mentioned [2], we represent the probability of the equality
x1 + x2 + + xn = m
for any given values of n and m as a sum ( Pm, n + Pm, n + Pm, n ) of the probabilities
Pm, n , Pm, n , Pm, n
of the same equality taken under an additional condition of xn = 1, 0, 1. Applying this
notation, it is not difficult to establish the following equations
Pm, n +1 = pPm +1, n + p1Pm +1, n + p2 Pm +1, n ; Pm, n +1 = qPm, n + q1Pm, n + q2 Pm, n ;
= rP
+ rP
+ r P
m , n +1
m 1, n
1 m 1, n
where n = 1, 2, 3,
We can make use of these equations for consecutively calculating the probabilities (1.7) at
n = 2, 3, 4, ; it is only necessary to take into account the main equalities
P1,1 = P, P0,1 = Q, P1,1 = R
and to note that all the other expressions Pm,1 , Pm,1 , Pm,1 , which are the probabilities of
impossible events, must be zero. We thus find that
P 2, 2 = pP, P1, 2 = p1Q, P0, 2 = p2 R; P1, 2 = qP, P0, 2 = q1Q, P1, 2 = q2 R;
P0, 2 = rP, P1, 2 = r1Q, P2, 2 = r2 R
whereas the other expressions, Pm, 2 , Pm, 2 , Pm, 2 , are zero. We obtain then
P 3,3 = ppP, P 2,3 = pp1Q + p1qP, P 2,3 = pqP, P 2,3 = 0, ...
P + P + P = P(pq + qp1) + Qp1p, etc.
Without considering the case in which
q2 = 0,
we may introduce in addition three numbers, P0,0 , P0,0 , P0,0 , determining them by the
P = pP0,0 + p1P0,0 + p2 P0,0 ; Q = qP0,0 + q1P0,0 + q2 P0,0 ; R = rP0,0 + r1P0,0 + r2 P0,0
on whose strength we have P0,0 + P0,0 + P0,0 = 1 .
At the same time, assigning a zero value to the symbols Pm,0 , Pm,0 , Pm,0 , m ! 0,
we may extend the deduced equations onto the case n = 0 as well.
2. Introducing a supplementary variable t and its functions
Pm, n t m , n =
Pm, n t m , n = n + n + n
we can, first, determine the probability of the equality (1.6) as the coefficient of t
development of n in powers of t; and, second, derive
t n +1 = p n + p1 n + p2n ; n +1 = q n + q1 n + q2n ;
(1 / t ) n +1 = r n + r1 n + r2 n
from the equations (1.8).
From (2.2) follows one and the same linear finite equation with an unknown function
for all the functions n , n , n , n . We can rather simply represent it in a symbolic manner
p t
r2 (1 / t )
where, after performing the operations indicated, n+3, n+2, n+1, n, should be replaced by
n+3,
n and n = 0, 1, 2,
We note the symbolic nature of the equation (2.3) because it clearly reveals how our
conclusions are extended onto other, more general cases in which xk can take more than three
different values. In an ordinary manner this equation is represented as
n+2 +
A = (p/t)+ q1 + r2t, B = [(pq1 p1q)/t] + pr2 p2r + (q1r2 q2r1)t,
D = pq1r2 pq2r1 + p1q2r p1qr2 + p2qr1 p2q1r.
On the other hand, in accord with our notation we have
cases n = 1 and n = 2 provides
= 1 and a direct study of the
= (P/t) + Q + Rt, 2 = (Pp/t2) + [(Pq + Qp1)/t] + Pr + Qq1+ Rp2 +
(Qr1 + Rq2)t + Rr2t2.
Issuing from these, we can determine, on the grounds of equation (2.3), all the other
functions n. And it is not difficult to see that all of them can be determined as coefficients
of zn in the expansion of some rational function
f(t; z)/F(t; z) in increasing powers of this new supplementary variable z. Here,
pz t
F(t; z) =
p1z
p2 z
= 1 Az + Bz2 Dz3
q1 z 1
q2 z
r1z
r2 z (1 / t )
and f(t; z) is an integral function of the second degree with respect to z; it is determined by
the first three terms of the formula
f(t; z)/F(t; z) =
A simple multiplication of the series just above by F(t; z) provides
+(
0)z
+B
o)z
For our purpose it is important to note that at t = 1 all functions
and that consequently
f(1; z)/F(1; z) = [1/(1 z)].
must reduce to unity
3. We shall apply the deduced formulas to calculate the expectations of various powers of
x1 + x2 + + xn na
and we shall select a in such a way that the expectation of the first power of this sum remains
finite as n
In accord with the definition of the function n, its coefficient of any given power of t is
the probability that the sum (1.5) is equal to the pertinent exponent; therefore, the probability
that the sum (3.1) is equal to a given number can be determined by expanding the product
in powers of t. In this expansion, the probability sought is the coefficient of the power of t
whose pertinent exponent is equal to the given number.
It is not difficult to conclude now that the expectation of
(x1 + x2 + + xn na),
where i is any integral positive number, can be obtained by making use of the product (3.2)
in the following way. Assume that t = eu, compose the derivative
d i[tna n]/ du i and calculate it at u = 0. The result thus obtained will provide the expectation
sought; the transfer from t to eu was necessary for the exponents to persist under
differentiation.
At the same time, it is not difficult to see that the series
+ ta
+t2a
+ + tna
is obtained from the series in (2.6) after replacing z by the product taz. It follows, on the
grounds of formula (2.6), that
f(t; zta)/F(t; zta) =
and the expectation of (3.3) might be determined as the coefficient of zn in the expansion of
d i{[f(eu; zeau)/F(eu; zeau)]/du i}u=0
in increasing powers of z.
4. Let us dwell now on the expectation of the first power of (3.1) so as to determine a.
According to our conclusion, this expectation is expressed by the coefficient of zn in the
development of
f u= 0 ( eu ; ze au )
f (1; z ) Fu= 0 (eu ; ze au )
F (1; z )
in increasing powers of z. Symbols f u=0 and F u=0 denote the values of derivatives at u = 0.
In order to study the expansion of the indicated function in increasing powers of z we shall
first of all consider its development into partial fractions; for our purpose it is important to
isolate only one fraction with denominator
(1 z)l. Obviously, only simple factors of the integral function F(1; z) can be the
denominators of the partial fractions sought.
One of its factors is (1 z) since its value at z = 1 is represented by the determinant
q1 1 q2 ,
which, on the strength of the main conditions (1.3), is zero. Turning now to the other simple
factors of the function F(1; z), we suppose that
F(1; z) = (1 z) (1 y1z) (1 y2z).
The numbers 1, y1 and y2 are the three roots of the equation
p y
q1 y
q2 = 0.
r2 y
We shall prove that its root y = 1 is simple, and that the moduli of the other roots are less
than unity. For this purpose, we compose the derivative with respect to y of the left side of
the equation (4.3) and assume that y = 1. The result obtained can be represented as a sum of
three differences
[q2r1 (1 q1) (1 r2)] + [p2r (1 p) (1 r2)] + [p1q (1 p) (1 q1)]
none of which can be positive and which can only vanish all at the same time in the cases
excluded by us, when at least one of the numbers p, q1, r2 is unity.
Ascertaining thus that unity is a simple rather than a multiple root of the equation (4.3), we
go over to the other roots, y1 and y2, of the same equation which differ from unity. Let y be
one of these other roots, it does not matter which. It is possible to select a system of numbers
, , ", not all of them being zeros, satisfying the equations
y = p + q +r", y = p1 + q1 + r1", "y = p2 + q2 + r2".
All the differences ( ), ( "), ( ") cannot vanish either, otherwise the equations (4.4)
lead to y = 1.
Consider the difference having the maximal modulus; let it be ( ). Accordingly,
subtracting y from y provides us with the following equality
( )y = (p p1) + (q q1) + (r r1)".
The absolute values of the coefficients (p p1), (q q1), and (r r1) of , , " are less than
unity; and, since their sum is zero, one of them has a sign opposite to the sign of the two
others, and is equal to their sum in absolute value. For the sake of definiteness let us assume
that (p p1) and (q q1) are of one sign, and (r r1) is of the other sign. Replacing now in
the equality above the difference (r r1) by an equal magnitude, the sum of the differences
(p1 p) and (q1 q), we have
y = [(p p1) ( ") +(q q1) ( ")]/( )
mod y | p p1| + | q q1| = | r r1| < 1.
Thus, the moduli of the coefficients y1 and y2, included in the development (4.2) of the
function F(1; z) in simple factors, are less than unity. Therefore, the coefficients of zn in the
well-known expansion of the fractions of the type
[1/(1 y1z) l], [1/(1 y2z) l]
in increasing powers of z must tend to zero together with (1/n).
As to the development of fractions of the type [1/(1 z) l] in increasing powers of z, we
[1/(1 z) l] = 1 + E1z + E2z2 + + Enzn + ,
where, on the strength of the formula
En = (n + 1)(n + 2) (n + l 1)/(l 1)!,
lim[En/n l1] = [1/(l 1)!], lim[En/n l1+ ] = 0, n
where is any given positive number.
Applying our conclusions to function (4.1), we infer that in its expansion in increasing
powers of z the coefficient of zn, equal to the expectation of (3.1), increases unboundedly
with n, if, and only if, (1 z) cannot be cancelled out of the second term of the expression
(4.1).
Desiring that this expectation will not increase unboundedly with n, and taking into
account that, on the strength of the equality (2.7), the integral function f(1; z) cannot include
the factor (1 z), we must assign to a such a value that this factor will be included in the
function F3u=0 (eu; zeau).
We thus arrive at the equation
F3u=0 (eu; zeau) = 0
that can easily be reduced to the following form
a[dF(1; z)/dz] z=1 + [dF(t; 1)/dt] t=1 = 0.
The coefficients of this equation, which admits of one and only one solution, can be
calculated by means of the formula
[dF(1; z)/dz] z=1 =
[q2r1 (1 q1) (1 r2)] + [p2r (1 p) (1 r2)] + [p1q (1 p) (1 q1)],
[dF(t; 1)/dt] t=1 =
(1 q1) (1 r2) q2r1 (1 p) (1 q1) + p1q.
5. We go on to consider the higher powers of the sum (3.1) with the determined value of a.
In accord with the above, the expectation of (3.3) can be derived as the coefficient of zn in the
expansion of the function (3.5) in increasing powers of an arbitrary number z.
Proceeding to study this function, we set, for the sake of brevity,
f(eu; zeau) = U, F(eu; zeau) = V, d iU/du i = U(i), d iV/du i = V(i).
In this notation we have, according to the formula for differentiating a product,
d i(U/V)/du i =[Ud i(1/V) du i] + (i/1) U3d i1(1/V)/du i1 +
and, making use of the formula for differentiating a function of a function, we obtain
d (1/V)/du =
i! j! ( V ) [(1 / 2)V ] [(1 / 6)V ]
V j +1 !
where the summation should extend over all the possible totalities of integral numbers
j, , , #,
satisfying two equations
+ + # + = j, + 2 + 3# + = i
and inequalities
0 < j i,
0, 0, # 0,
We want to prove that, as n
)C i/2
t iexp( t2) dt
, the ratio of expectation of (3.3) to n tends to
where C is some constant. For this purpose, bearing in mind the conclusions above, it is
necessary to ascertain that the denominator of the rational function of z, represented by the
value of the derivative d i(U/V)/du i at u = 0 can include factor (1 z), of course, after
appropriate cancellations, only in a power not higher than (i + 1)/2 for odd values of i, and
not higher than [(i/2) + 1] for its even values.
Considering one of the products included in formula (5.2)
[U (l)d il(1/V)/du il]u=0,
we note that the factor (1 z) cannot be present in the denominator of this product in a power
higher than in the denominator of its factor
[d il(1/V)/du il]u=0.
Concerning this second factor and issuing from the formula (5.3) with number i replaced
by (i 1), we can, however, derive important conclusions. Indeed, it is not difficult to see
that, with the selected value of a, the general term of formula (5.3) taken at u = 0 is such an
irreducible fraction whose denominator includes the multiplier (1 z) in a power not higher
than (j + 1 ) because the function V at u = 0 includes this multiplier whereas the function
V is not divisible by (1 z)2. On the other hand, issuing from the conditions imposed on the
magnitudes (5.4), it is not difficult to derive the inequality
2j i
that restricts the values of .
Until 2j < i, the difference
where 0 obviously remains less than [(i/2) + 1]. On the strength of inequalities (5.7) this
restriction persists also when 2j > i. And only if j = i/2, if such value is possible, and = 0, it
can attain the value [(i/2) + 1]. Dwelling on the supposition
i = 2j, = 0,
which is possible only for even values of i, we find that only one term of the formula (5.3),
[i!/2i/2]( V ) i/2/V i/2+1
corresponds to this assumption. Indeed, equations (5.5) and their associated inequalities then
lead to = 1/2.
It follows that for odd values of i not a single fraction, to which the derivatives
[d i(1/V)/du i], [d i1(1/V)/du i1], [d i2(1/V)/du i2],
are reduced if u = 0, can include in its denominator, of course after appropriate cancellations,
the factor (1 z) in a power higher than (i + 1)/2; for even values of i only the first of these
fractions obtained from the derivative [d i(1/V)/du i] can include in its denominator the factor
(1 z) in the power [(i/2) + 1]. If, however, the expression (5.9) at u = 0 is subtracted from
this first fraction, the difference, after appropriate cancellations, is reduced to a fraction
whose denominator includes the factor (1 z) in a power lower than
[(i/2) + 1].
Comparing this result with formula (5.2) and taking into account notation (5.1), we
conclude that the expression studied (3.5) is reduced, for odd values of i, to such a rational
fractional function of z whose denominator includes the factor (1 z) in a power lower then
[(i/2) + 1]. It immediately follows, on the strength of the investigations in 3 and 4, that for
odd values of i the ratio
[E(x1 + x2 + + xn na)i]/n i/2
must tend to zero as n
. Neither is it difficult to indicate the partial fraction with
denominator (1 z) i/2+1 which should be isolated from the expression (3.5) so that the
denominator of the residual will include only lower powers of (1 z), and not (1 z) i/2+1
Indeed, this fraction must coincide with the one whose isolation from the expression
[i!/2i/2] (U/V)u=0 ( V /V) i/2u=0
leaves a fraction that does not include the multiplier (1 z) in a power higher than [(i/2) 1]
in its denominator. Therefore, taking into account the equality (2.7), we find that the fraction
sought can be represented as
[i!/2i/2] [C i/2/(1 z) i/2+1].
Here, C is determined by the formula
C/2 = F2u=0 (eu; eau)/F z=1(1; z).
After arriving at such a conclusion and comparing it with the investigations of 3 and 4,
we satisfy ourselves without any serious additional work that for even values of i the ratio
(5.10) tends, as n 0 1, to
C i/2.
2 (i / 2)!
It remains to take into consideration that the integral (5.6) vanishes if i is odd and is equal
otherwise, and we arrive at the final conclusion. If a is determined by the equation (4.5) and
C is given by the formula (5.11), then, both for odd and even values of i, as n ,
lim[E(x1 + x2 + + xn na)i]/n i/2 = (1/
t iexp( t2) dt.
Consequently, as n
, the probability of the inequalities
t1 Cn < x1 + x2 + + xn na < t2 Cn
where t1 and t2 are any given numbers with t2 > t1, must tend to
exp( t2) dt.
6. After considering a case of quantities connected into a chain and noting that it does not
present any peculiarities but differs from the other ones only in the simplicity of the initial
data, we may in a few words extend our deductions onto the general instance of such
quantities studied in the abovementioned article [3].
Keeping to the notation in this paper, suppose that
are the various possible values of the numbers in the chain
x1, x2, , xk, xk+1, ,
and that the system
p , p , p ", ; p , p , p ", ; p" , p" , p"", , etc
represents the probabilities, given the values of xk, for xk+1 to take definite values; the
subscripts of the ps indicate the given xk and the supposed xk+1, respectively.
The numbers (6.1) and (6.3) determine our final conclusions, but, to pose a quite definite
problem, we ought to introduce additionally numbers
p3 , p3 , p3",
that, just as the numbers (1.4) of the particular case did, fail to appear in the final
conclusions, and are the probabilities of the equalities
x1 = , x2 = , x3 = ",
respectively as long as all the xs of our chain remain indefinite.
Turning to considering various assumptions about the value of the sum (1.5), we denote
the probability of the equality (1.6) by symbol Pmn and introduce a function n of an
arbitrary number t
= Pmn t m.
Acting now in the same way as we did in the particular case, we partition Pmn:
Pmn = Pmn, + Pmn, + Pmn," +
and introduce a number of functions
= Pmn, t m,
where the new symbols are the probabilities of the same equality (1.6) and of an additional
restriction expressed by one of the equalities xn = ; ; ";
Having this notation at our disposal, it is not difficult, in accord with the conditions of the
problem, to establish the equations
Pmn, = Pm
,n1,
p +Pm
p + ,
then passing on from them to equations
t = p
n1,
+ p"
n1,"
One and the same homogeneous difference equation follows now for all the functions n, ,
n, ,
n,", as well as for
n, +
n," + It can be symbolically represented in
a rather simple way as
p t ...
where, after performing the indicated operations, n, n+1, n+2, should be replaced by
n+2, respectively.
On the strength of this equation all the functions 0, 1, 2, , n, can be determined
as coefficients in the expansion of some rational function
f(t; z)/F(t; z) of a second
supplementary number z in its increasing powers. The denominator of this function is
determined as
p z t
p z
Before going on to further conclusions, it is necessary to note that we are only considering
such chains (6.2) for which the appearance of some numbers (6.1) does not definitively
exclude the possibility of the appearance of other numbers. This important condition can be
expressed in the following way: the determinant
with arbitrary elements u, v, w, is not reducible to a product of several determinants of the
same type.
This, however, is not sufficient for our purpose, and we ought to suppose 1 that the
determinant above cannot be explicitly reduced to a product of several determinants when u
= p , v = p , w = p"", either.
Both in the particular and in the general case our conclusions concern the expectation of
the power (3.3) where a is determined by demanding that at i = 1 this expectation will not
increase unboundedly with n. Having established, as before, that it can be expressed by the
coefficient of zn in the expansion of the value of the derivative (3.5) in increasing powers of
z, we note that, for extending the inferences made in the particular case onto the general
instance, we ought to consider the development of the function F(1; z) in linear factors
F(1; z) = (1 z) (1 y1z) (1 y2z)
and to prove that the multiplier (1 z) occurs here only once and that the moduli of y1, y2,
are less than unity.
In other words, we ought to satisfy ourselves that unity is a simple rather than a multiple
root of the equation
p y ...
and that the moduli of the other roots of the same equation, y1, y2, , are less than unity.
The following proposition regarding determinants can serve to prove the first point. If all
the elements of the determinant
c1 ...
w ...
satisfy inequalities
ak 0, bk 0, ck 0,
u a1 + a2 + , v b1 + b2 + , w c1 + c2 + ,
it cannot be negative; and it can vanish only in the extreme case in which all the inequalities
(6.14) are reduced to equalities, or in which it becomes, owing to the reduction of some of
the inequalities (6.13) to equalities, a product of several determinants of the same type with,
moreover, one of the latter being such that all the inequalities similar to (6.14) are reduced to
equalities 2.
We become convinced in this important proposition when considering u, v, w, as
variables and noting that the derivatives of our determinant with respect to them are
expressed by similar determinants of a lower order; such a reasoning enables us to extend
gradually this theorem from a determinant of the second order, for which it is evident, onto a
determinant of the third; then to one of the fourth order, etc.
It immediately follows from the proposition thus proved that under our conditions the
derivative of the left side of equation (6.12) with respect to y does not vanish at y = 1. Indeed,
after being multiplied by 1, it can be represented as a sum of the determinants
... + p
... +
obeying the conditions of the theorem and not presenting the extreme case.
Thus, unity, which on the strength of the conditions
p + p + p " + = 1, p + p + p " + = 1,
must satisfy equation (6.12), cannot be its multiple root. Turning to the other roots of this
equation, we assume that y is any one of them. A system of numbers
, ,",
not all of them zeros and satisfying the equations
y=p
" y = p"
+ p "" + , y = p
+ p""" + ,
+ p "" + ,
can be chosen accordingly. These numbers cannot coincide because the equations above will
then lead to y = 1. Taking this into account, we suppose at first that the numbers (6.15) do
not have a common modulus either. Then, owing to our conditions, among the equations to
which the numbers (6.15) are subordinated, there must be at least one, the coefficient of y in
whose left side will be one of these numbers with the greatest modulus and whose right side
will also include, with a non-zero coefficient, another one of these numbers (6.15) with a
lesser modulus. It follows immediately, since the sum of the coefficients of the numbers
(6.15) in the right sides of each of these equations is unity, that
mod y < 1.
Let us suppose now that all the numbers (6.15) have the same modulus. We know,
however, that not all of them are equal one to another. Their totality can therefore be
partitioned in two groups with respect to the first number, : the numbers in the first group
being equal, and in the second group, unequal to . The numbers in the second group differ
from in their arguments. On the other hand, because of one of our main conditions, it is
impossible to partition the totality of the sums
+ p "" + ; p
+ p "" + ;
in two totalities in such a way that the first sums will include, with non-zero coefficients 3,
only the numbers equal to and the second sums will include only the other numbers.
Therefore, one of these sums will certainly include, with non-zero coefficients, numbers
both equal and unequal to , and its modulus, equal to the product (mod y) (mod ), must be
less than mod since for numbers with differing arguments the modulus of their sum is less
than, and not equal to the sum of their moduli. The inequality (6.16), which we should have
proved, follows now immediately.
Having thus proved that in developing the function F(1; z) in simple multipliers (6.11) the
factor (1 z) only enters in the first power, and that the moduli of the coefficients y1, y2,
are less than unity, we may transfer without change all the further deliberations, that we have
applied to the particular instance, onto the general case. Since our reasoning remains entirely
invariable, we shall not repeat it and only adduce the final conclusion.
If, under the stipulated conditions and notation, we define numbers a and C by equalities
{cf. (5.11)}
a = F t=1(t; 1) / F z=1(1; z), C/2 = F2u=0 (eu; eau)/F z=1(1; z),
which does not lead either to impossible or indeterminate results, then, for any integral
positive i {see end of 5}, as n ,
and, consequently, as n 0 1, the probability of the inequalities
It can also be noted about the numbers a and C that, as n 0 1, they are equal to the limits
[E(x1 + x2 + + xn)/n ], 2[E(x1 + x2 + + xn na)2/n]
respectively. This remark allows to extend our conclusions also onto such cases in which the
totality (6.1) is not exhausted by a finite number of terms. It is also possible to extend our
conclusions onto generalized chains whose each term is directly connected not with one, but
with several preceding terms.
1. Our conclusions may be extended onto many of the excluded cases.
2. A similar proposition is in Minkovskys note [6].
3. Previously, I [3] have only considered only the simplest assumption that among these
coefficients none is equal to zero.
{In several works} Markov proves the applicability of the CLT to sums
of random variables Xi []. {Here,} he studies variables Xi forming a simple homogeneous
chain and especially dwells on the assumption that the possible values of the variables are
1, 0, and 1. To calculate the moments of Sn, he introduces generating functions
n(t)
pmnt m
with pmn being the probabilities of the equalities Sn = m, and studies the generating function
of the functions n(t)
zn = f(t; z)/F(t; z).
The main role in the ensuing calculations was played by the denominator of this fraction,
F(t; z), which, as it appeared, was a polynomial of the third degree with respect to z. Under
some restrictions of a general nature concerning transition probabilities, it turned out that
F(1; z) had a simple root z = 1 and two other ones whose moduli were less than 1. On the
contrary, the function f(t; z) did not play an important part nor did it influence the results of
calculation as n 0 .
Concerning the moments E[(Sn an)/ n]i where a was an adequately chosen number,
Markov showed by investigating the right side of (1), that, for any integral i = 0, 1, 2, and
n0 ,
E[(Sn an)/ n] 0 (1/
)C
i/2
C = [2 2F(eu; eau)/ u2] u=0 / F(1; z)/ z z=1.
The proof was largely based on the fact that z = 1 was a simple, and, furthermore, the only
root of the function F(1; z) having maximal modulus. A similar reasoning leading under
these conditions to (2) can be reproduced for many other cases concerning the variables Sn
In concluding his work, in 6, Markov studied variables Xi with values
He proved that the CLT was applicable to Sn under conditions, which, in our opinion, were
not formulated distinctly enough. He said that he was considering only such chains for which
the appearance of {certain} numbers (3) [did] not exclude definitively the possibility of the
appearance of other numbers.
In other words, using modern terminology, he studied homogeneous chains whose
matrixes of transition probabilities (stochastic matrixes) were simple [1, Addendum 5]; or, all
the states of the chain formed one class of essential states [2]. Then Markov put forward this
condition: The determinant (6.10) with arbitrary elements u, v, w, is not reducible to a
product of several determinants of the same type, and, also, the determinant above cannot be
explicitly reduced to a product of several determinants when u = p , v = p , w = p"",
either. He then stated that under these conditions the characteristic equation (6.12) has only
one root of mod y = 1, namely y = 1 whereas all the other ones have moduli less than 1. It is
known that in the case of a simple stochastic matrix a necessary and sufficient condition for
the equation (6.12) to have only one root, maximal in modulus, namely, y = 1, is, in
Bernsteins terminology, the regularity of this matrix; in other words, its period must be
equal to 1. []
In concluding, we note that, for being irreproachable, Markovs inferences about the
applicability of the CLT to variables Sn should be supplemented by demanding that the nonnegative constant C in equality (6.17) [] be positive.
(Theory of Probability). M. L. Fourth
2. Kolmogorov, A.N. (1937), Markov chains with a countable set of possible states. Bull.
Moskovsk. Gos. Univ., vol. 1, No. 3. (R)
3. Markov, A.A. (1906), Extension of the law of large numbers, etc. Translated in this
4. --- (1907, in Russian), Recherches sur un cas remarquable dpreuves dpendantes. Acta
math., t. 33, 1910, pp. 87 104.
6. Minkowski, H. (1900), Zur Theorie der Einheiten in den algebraischen Zahlkrpern.
Nachr. Kgl. Ges. Wiss. Gttingen, math.-phys. Kl., 1900, pp. 90 93.
13. A.A. Markov. An Investigation of the General Case
of Trials Connected into a Chain
When investigating the simplest case of trials connected into a chain, we [5] have
ascertained the possibility of extending the LLN onto it. Then, in addition, we [6] established
for it the theorem on the limit of expectation, from which, on the strength of the works of
Chebyshev and myself, follows the theorem on the limit of probability. Our conclusions were
based, in their most difficult stage, on the use of generating functions, which, for that case,
were expressed rather simply. Their consideration also allowed us to extend the same
inferences onto any homogeneous 1 chain of magnitudes [7].
Owing to the complexity of the generating functions themselves, this method will hardly
simplify the investigation of non-homogeneous chains. Therefore, desiring to extend the
theorems on the limits of expectation and probability onto magnitudes connected in such
chains, we were compelled to return to those methods, which we had successfully applied to
various cases of independent magnitudes and which consisted in decomposing, on the
strength of the Newton formula, the expectation of the power of the sum into separate sums,
then isolating from them those sums, that increase more rapidly than any other ones, of
course with the increase in the number of magnitudes under consideration.
Here, we shall dwell on the general case of trials connected into a non-homogeneous chain
that does not contain such links where the probability of the event can happen to be either
zero or unity, or be arbitrarily close to these numbers. Note that the word probability, as was
already established [6], means here, for each trial, not one but three numbers corresponding
to different assumptions.
1. Beginning our investigations, we ought first of all to establish the relevant conditions
and introduce notation. We shall consider an unbounded series of consecutive trials
distinguishing them, in their order, by numbers 1, 2, 3, At each of these trials, either some
event E or its contrary event F can occur. Our trials are connected into a chain in such a way
that for any integral positive number k we have
1) The probability of event E at the (k + 1)-th trial, until the results are not at all known,
has a definite value pk+1; the probability of F under the same condition is qk+1 = 1 pk+1.
2) The probability of event E at the (k + 1)-th trial takes another definite value, p k+1 if the
results of the consequent trials are indefinite as before, but the directly preceding trial, i.e.,
the k-th trial, furnished event E, whatever were the results of the other ones, i.e., of the first,
the second, , and the
(k 1)-th trial. The probability of event F under these new
conditions is q k+1 = 1 p k+1.
3) Finally, the probability of E at the (k + 1)-th trial takes a third definitive value, p k+1 if
the results of the consequent trials remain indefinite as before, but the directly preceding trial,
i.e., the k-th trial, had not provided event E whatever were the results of the other ones. The
probability of event F in this last case is q k+1 = 1 p k+1.
On the strength of these conditions, trial (k + 1) is only connected with the previous ones
through the k-th trial and becomes independent from them once the last-mentioned one is
determined. We must complement the data indicated above by the probabilities p1 and q1 = 1
p1 of the events E and F at the first trial supposing that the results of all of the trials remain
p1, p 2, p 2, p 3, p 3,
may be assigned arbitrarily; and, issuing from them, it is not difficult to calculate
consecutively p2, p3, , pk, pk+1, by means of a simple general formula
pk+1 = pk p k+1, + qk p
k+1.
This formula can also serve for another purpose: by its means, we can express p k+1, p
q k+1, q k+1 through
pk+1, qk+1, pk, qk and
= p k+1 p
k+1,
Indeed, denoting
as above, we derive from formula (1.1)
pk+1 = p k+1 (pk + qk)
qk = p k+1
qk
and can therefore immediately establish a number of simple equalities
p3k+1 = pk+1 +
p2k+1 = pk+1
qk, q k+1 = qk+1
pk, q k+1 = qk+1
qk,
pk.
We may thus introduce into our calculations pk+1, qk+1, k+1 instead of p k+1, p k+1, q k+1 and
q k+1 and we shall indeed do it so as to simplify our deductions as much as possible. It should
be noted that p k+1 and p k+1 are only restricted by inequalities 0 p k+1 1 and 0 p2k+1
1 so that 1 k+1 1. If, however, we introduce pk+1 and k+1 instead of p k+1 and p2k+1,
then, as can easily be ascertained, the numbers k+1 ought to be restricted by the new
(pk+1/qk)
(qk+1/qk), (qk+1/pk)
(pk+1/pk),
whereas the numbers pk+1 must only obey inequalities 0 pk+1 1.
2. Let us dwell now on a problem, important for our aim, of calculating the probability of
event E at the k-th trial, at first under the condition that it had occurred at the i-th trial (i < k),
and then under the contrary condition. Denoting for this purpose the former probability by
Pk(i), and the latter by Qk(i), and considering different values of k, we deduce simple equations
Pk+1(i) = Pk(i) p k+1 + (1 Pk(i)) p k+1 = pk+1
Qk+1(i) = Qk(i) p k+1 + (1 Qk(i)) p k+1 = pk+1
Pk(i),
k+1 Qk .
pk +
k+1 pk +
At the same time we note, that, because of our notation,
Pi+1(i) = p i+1 = pi+1 +
i+1
qi, Qi+1(i) = p
= pi+1
Issuing from these equalities, it is not difficult to deduce consecutively
Pi+2(i) = pi+2 +
qi, Qi+2(i) = pi+2 i+1
i+2 i+3 qi, Qi+3 = pi+3
i+2 pi,
and, in general, to arrive at
Pk(i) = pk +
qi, Qk(i) = pk
k pi.
Thus, under the stated condition, the probability of the event E at the k-th trial is
determined when the result of the i-th trial (i < k) is known.
Neither is it difficult to establish the formulas for calculating the probabilities Pk(i) and Qk(i)
of event E at the i-th trial under the condition that E took place, or alternatively, that it did
not occur at the k-th trial (as before, i < k). These last-mentioned formulas directly follow
from formula (2.1) and from the simple equalities
piPk(i) = pkPi(k), pi(1 Pk(i)) = qkQi(k).
Indeed, we have
Pi(k) = pi +
piqi,/pk, Qi(k) = pi +
piqi,/qk.
Concerning formulas (2.1) and (2.2), we note that, in accord with what we said in the
introduction, neither of the numbers pk or qk can be zero or arbitrarily close to it; therefore,
1 > p > pk > p > 0
hold for some constant magnitudes p and p and any subscript k. Introducing a similar
assumption concerning all the numbers p k+1 and p k+1, we ought also to establish the general
1< <
< <1
where is the same for all values of k. These inequalities will play an important part in our
conclusions about the limiting values of the studied expressions.
It is of course possible to pose the question on the extension of the final deductions also on
such cases, for which some of the numbers pk, 1 pk, and k+1 are arbitrarily close to unity.
However, here we shall not consider this difficult problem leaving it for other researchers.
On the contrary, not wishing to ascertain how rapidly is one rather involved expression
increasing, we believe it reasonable to restrict, in this paper, the numbers k+1 by some
additional inequalities; being complied with, they ensure that the investigation of this
problem presents no difficulty. We shall introduce these inequalities at an appropriate time;
for the present, those established above are sufficient.
They are sufficient, for example, for deriving from formulas (2.1) and (2.2) the inequalities
|Pk(i) pk| <
|Pi(k) pi| <
, |Qk(i) pk| < ki,
/4p, |Qi(k) pi| <
/[4(1 p)]
on whose strength we can infer that all these differences tend to zero when (k i)
However, this last conclusion can of course be also justified in many other cases on which
we do not dwell.
3. Turning now to consider the totality of the first n trials, denote the number of the
occurrences of event E in them by m, and, in accord with a well-known rule, let
m = x1 + x2 + + xk + + xn.
Here, xk is unity if event E occurs at trial k and zero otherwise, when at this trial happens
event F. We shall study the possible values of m assuming that the results of all the trials
remain indefinite.
Under this assumption we shall consider the expectations of various powers of the
m (p1 + p2 + + pn)
equal to the sum
(x1 p1) + (x2 p2) + + (xn pn)
and compare these expectations with powers of n. We attempt to isolate the main part of each
expectation which determines the law of its increase as n
The expectation of the first power of the difference (3.2) is obviously zero since
Ex1 = p1, Ex2 = p2, , Exn = pn.
Neither is it difficult to compile the expectation of its square. To achieve this we make use of
the well-known equality
[(x1 p1) + (x2 p2) + + (xn pn)]2 = (xk pk)2 + 2 (xi pi) (xk pk)
with k = 1, 2, , n in the first sum, and i = 1, 2, , n 1 and k = i + 1, , n in the second
one. We then calculate the expectations of the squares (xk pk)2 and the products (xi pi) (xk
pk). Applying formulas (2.1), we deduce
E(xk pk)2 = pk qk2 + qk pk2 = pk qk,
E(xi pi) (xk pk) = pi qi (Pk(i) pk) qi pi (Qk(i) pk) =
pi qi (Pk(i) Qk(i)) = i+1 i+2 k pi qi.
Therefore, for i = 1, 2, , n,
E[m (p1 + p2 + + pn)]2 = pi qi (1 + 2
p1 q1 (1 + 2 2 + 2 2 3 + 2
pn1 q n1 (1 + 2 n) + pn q n.
i+1 i+2
n ++
Our expression for the expectation of the square of (3.2) enables us to extend the LLN in a
very simple way onto this case (onto the Bernoulli theorem) if only all the | i+1| remain less
than one and the same number , which, in accord with our abovementioned condition, is
itself less than unity. Indeed, if
< , <
< , , <
< < 1,
not a single sum from among
1 + 2 n, 1 + 2
n-1 n,
, 1 + 2
++ 2
attains the value
[(1 + )/(1 )] = 1 + 2 + 2
so that the formula (3.4) provides the inequality
E[m (p1 + p2 + + pn)]2 < n (1 + )/[4(1 )].
The LLN is derived from this inequality by means of a well-known reasoning; therefore,
with probability arbitrarily close to certainty, we may, for a sufficiently large n, state that the
[m/n] [(p1 + p2 + + pn)/n]
is less in absolute value than any given number.
For our further deliberations concerning the expectations of the higher powers of the
difference (3.2), it is important that the ratio
E[(x1 p1) + (x2 p2) + + (xn pn)]2/n
will not become arbitrarily small. We do not dwell on the question of whether the inequalities
indicated above are sufficient for the last requirement to be fulfilled; we shall only indicate
several cases in which it is not difficult to ascertain that this condition is indeed obeyed. First,
if all the magnitudes
are positive, then each sum (3.5) is not less than unity, so that, with the general inequalities
(2.3) persisting, formula (3.4) leads to a simple inequality
{E[(x1 p1) + (x2 p2) + + (xn pn)]2/n} > p (1 p).
If, however, some of the numbers (3.7) are negative, but , introduced earlier, is less than
1/3, then each sum (3.5) is larger than the positive number
[(1 3 )/(1 )] = 1 2 2
2 3 ,
and, owing to formula (3.4), we can also establish a simple inequality: the left side of (3.8) is
(1 3 ) p (1 p)/(1 ).
Having thus ascertained the existence of cases in which the ratio (3.6) cannot be arbitrarily
small, we shall assume further on that this condition is fulfilled, leaving room for doubting
whether it represents a new restriction, or holds as a corollary of those established earlier 2.
4. Before considering the expectation of
[(x1 p1) + (x2 p2) + + (xn pn)]m
for any given integral positive number m, we shall dwell on the particular case of m = 3
whose investigation can serve for throwing light on some peculiarities of our forthcoming
reasoning. We must prove that the expression
x1 p1 + x2 p2 + ... + xn pn
E ( x1 p1 + x2 p2 + ... + xn pn )2
for the ratio
. For this purpose, we note that it must tend to zero if the same is true
E[(x1 p1) + (x2 p2) + + (xn pn)]3/n3/2
which, in accord with the condition formulated at the end of 3, only differs from (4.1) by a
finite factor.
Then we pay attention to the well-known simple equality
[(x1 p1) + (x2 p2) + + (xn pn)]3 = (xi pi)3 + 3 (xi pi)2(xk pk) +
3 (xi pi)(xk pk)2 + 6 (xi pi)(xk pk)(xr pr)
1 i n; 1 i < k n; 1 i < k n; and 1 i < k < r n,
respectively. The equality follows as a particular case from the generalized Newton formula
with respect to a power of a sum. Consequently, we consider the expectations of the four
sums.
The expectation of the first of these is equal to the sum pi qi(qi2 pi2) consisting of n
terms not one of them attaining 1/4 in absolute value; therefore, the ratio E ( xi pi)3/n
remains finite, and the ratio E ( xi pi)3/n3/2 tends to zero as n
. Turning to the second
sum, we find that
E[(xi pi)2(xk pk)] = piqi2 (Pk(i) pk) + qipi2(Qk(i) pk) =
piqi (qi pi) i+1 i+2 k
E[(xi pi)2(xk pk)] = p1q1(q1 p1)[ 2 + 2 3 + + 2 3 n] +
p2q2(q2 p2)[ 3 + 3 4 + + 3 4 n] + + pn1qn1 (qn1 pn1) n.
This equality reveals that the ratio
E [(xi pi)2(xk pk)]/n
also remains finite. Therefore
lim{E [(xi pi)2(xk pk)]/n3/2} = 0 as n 01
since all the products p1q1(q1 p1), p2q2(q2 p2), , pn1qn1 (qn1 pn1) are less than 1/4 in
absolute value and any sum
i+3 +
remains under our conditions less than /(1 ).
In a similar way we obtain for the third sum
E[(xi pi)(xk pk) 2] = piqi[Pk(i)qk2 + (1 Pk(i))pk2 Qk(i)qk2 (1 Qk(i))pk2] =
piqi(qk pk) i+1 i+2 k
and easily conclude that (4.3) is indeed valid.
It remains to consider the expectation of the last sum which consists of the products (xi
pi)(xk pk)(xr pr) with i < k < r. The expression for the expectation of such a product can be
obtained by various means. We shall base our calculations on the fact that xi and xr become
independent one from another once the value of xk is determined. And two values, 0 and 1,
with probabilities pk and qk respectively, are possible for xk. At the same time, it is not
difficult to see that, when xk= 1, the expectations of xi and xr are equal to Pi(k) and Pr(k); or,
Qi(k) and Qr(k) respectively when xk= 0.
Therefore, taking into account formulas (2.1) and (2.2), we determine that, for i < k < r,
E[(xi pi)(xk pk)(xr pr)] =
pkqk[(Pi(k) pi) (Pr(k) pr) (Qi(k) pi) (Qr(k) pr)] =
piqi(qk pk) i+1 k k+1 r
E [(xi pi)(xk pk)(xr pr)] = piqi(qk pk)
k k+1
= piqiCi
where, respectively,
1 i < k < r n for the first two, and 1 i n for the third sum;
Ci = (qk pk) i+1 k
Dk = k+1 r = k+1 +
= (qk pk) i+1
k+1 k+2 n.
k+1 k+2 +
kDk (i
< k < r n),
And, consecutively considering Dk and Ci, we obtain simple inequalities
|Dk| < /(1 ), |Ci| < [ /(1 )]2
from which we immediately arrive at
[|E (xi pi)(xk pk)(xr pr)|/n] < (1/4)[ /(1 )]2
lim{E [(xi pi)(xk pk)(xr pr)]/n3/2} = 0 as n 0 1.
Thus, as n 0 1, all the four ratios of E (xi pi)3, E (xi pi)2(xk pk),
E (xi pi)(xk pk)2, and E (xi pi)(xk pk)(xr pr) to n3/2 tend to zero, and, together with
them, the ratio (4.2) must also tend to zero on the strength of the abovementioned simple
5. Finally, let us now pass over to general conclusions without dwelling separately on the
case of m = 4, 5, Issuing from the generalized Newton formula, we obtain the equality for
any given positive integer m
[(x1 p1) + (x2 p2) + + (xn pn)]m =
G (xi pi) (xj pj) (xr pr) , G = m!/( ! ! !)
and the summations are over , , ,
totalities of integral positive numbers
satisfying the condition
and i, j, k, , r respectively extending onto all the
++ =m
and onto the totalities of integral positive numbers
i, j, k, , r
obeying the inequalities
i < j < k< < r n.
It is important to remark that the number of the sums extended over i, j, k, , r in (5.1) does not
increase unboundedly with n but remains finite. Dwelling on one such sum, we shall consider
its expectation, i.e., the expectations of its terms
(xi pi) (xj pj) (xk pk)" (xr pr) .
The expectation of such a product is represented by a sum of a finite number of terms of the
( i pi) ( j pj) ( r pr) A(B + G
j) (C + H
Here, each of the numbers i, j, k, , r is equal to unity or zero, A is the probability of
the equality xi = i; the sum (B + G i+1 j) is the probability of the equality xj = j when the
former equality is already established, and the sum (C + H j+1 k) is the probability of the
equality xk = k when the equality xj = j is already established, etc. It is important to note that
all the numbers A, B, G, C, H, always remain finite since they are determined by the
A = pi or qi, B = pj or qj, G = pi or qi, C = pk or qk, H = pj or qj,
Neither is it difficult to represent the compiled expression for the expectation of the
product (5.6) as a sum
L + M1
N1 i+1
+ M2 j+1 j+2
j+1 k + +
arranged according to the products
k,
and to their various combinations into involved products. On the strength of the explanations
provided, we may state that, for any given system of exponents (5.2), all the magnitudes
L, M1, M2, , N1,
remain finite no matter how large n will be, even if n and whichever values the
subscripts (5.4) only obeying the inequalities (5.5) will take.
Pursuing a definite purpose, we shall not study the composition of the expressions (5.9);
we shall only indicate that the elimination from the sum (5.7) of all the terms containing a
definite product from our totality of products (5.8) is equivalent to severing the chain of
magnitudes
xi, xj, xk, , xr
at the respective place. Instead of one chain we shall thus obtain two chains, and the
expectation under consideration will be replaced by a product of two expectations.
If, however, only those terms that do not contain some of the products (5.8) are left in the
sum (5.7), the chain of the magnitudes (5.10) will be separated into several isolated chains,
and instead of the expectation under consideration we shall obtain a product of several
expectations. For example, when eliminating all the terms containing the factor j+1 k
from (5.7), the remaining sum will be equal to
E (xi pi) (xj pj) E(xk pk)" (xr pr) .
If, however, we only leave there those terms that do not contain either
we shall obtain, instead of the expectation considered,
E (xi pi) E(xj pj) E(xk pk)" (xr pr) .
Finally, the first term L, to which our sum is reduced after rejecting all those containing the
factors (5.8), is of course equal to the product of the expectations of the separate exponents,
E (xi pi) E(xj pj) E(xr pr) .
We shall make use of the deliberations about partitioning the chain into several separate
chains for determining the main term of the expression
E (xi pi) (xj pj) (xk pk)" (xr pr) ,
i.e., of that term that plays the decisive role in our investigation.
However, as a preliminary, we shall establish the formula for expectation (5.11): it is equal
L + M1 i+1
N1 i+1 i+2
k++
This follows from the expression (5.7) for the expectation of the product (5.6). The sums
extend over all possible values of the subscripts satisfying the inequalities (5.5).
We can accomplish the summing in several stages, by transforming it into multiple sums,
only extending it each time over one subscript. Arranging these separate operations in the
order of decreasing subscripts r > > k > j > i so as to compile the appropriate sum, we
must assign all possible values exceeding the next subscript but not exceeding n, to each
subscript occupying a definite place. The number of sums is equal to the number of
exponents (5.2).
Turning now to the problem of the possible rapidity of the increase in our multiple sums
comprising the formula (5.12) as n
, we may, with respect to each of them separately,
partition the just mentioned consecutive summings over the subscripts r,, k, j, i into two
groups in such a manner, that, as far as our purpose is concerned, all the operations in the
first one may be compared with repeating the same number n times, and the summings in the
second group, with a geometric progression with common ratio . And it is not difficult to
conclude from here that the ratio of the considered multiple sum to the power of n, equal to
the number of the appropriate sums in the first group, must remain finite as n
. In
particular, for L we should attribute all the summations over the separate subscripts to the
first group, and, consequently, in those cases in which we cannot establish that L = 0, we may
only maintain that the ratio L/n , where is the number of the exponents (5.2), must remain
finite.
For any other sum in (5.12) the number of operations in the first group is less than so
that its ratio to n must tend to zero as n
; for example, the sum
taken over the subscript j should be attributed to the second group and there will be left only
( 1) summings in the first one, and
taken either over j or over k should be attributed to the second group, so that only ( 2)
summings will be left in the first one.
We may therefore state that the difference
[E (xi pi) (xj pj) (xk pk)" (xr pr) /n ] L/n
. At the same time,
L = E(xi pi) E(xj pj) E(xk pk)" E(xr pr)
will always be the main term of the expression (5.12) excepting such cases in which some of
the exponents (5.2) are unity. In these instances, which we shall now consider, the sum L
vanishes since all of its terms are zero.
6. Dwelling now on the case in which some of the exponents (5.2) are unity, we introduce
for the sake of brevity the following expressions:
1) For subscripts i, j, k, , r satisfying inequalities (5.5) the products (5.8) will be the
coefficients of connection between i and j, j and k, respectively.
2) We shall call the terms of (5.2), appearing in the product (5.6), the exponents of the
subscripts (5.4).
These expressions allow us to formulate, concisely and clearly, the following proposition:
If the exponent of any of the subscripts (5.4), satisfying inequalities (5.5), in the product (5.6)
is unity, then each term of the expression (5.7) for the expectation of (5.6) contains at least
one coefficient of connection of this subscript with its adjacent subscripts. In particular, if =
1, expression (5.7) must include as a common multiplier the coefficient of connection
between i and the next subscript, j, equal to the first product in (5.8); for = 1, the common
multiplier will be the coefficient of connection between r and the directly preceding
subscript.
This proposition can be proved briefly and simply by means of our previous reasoning
concerning the partition of the chain (5.10) into several isolated chains. Indeed, after isolating
one subscript from among (5.4), and excluding all terms containing the coefficients of
connection of this subscript with its adjacent ones from expression (5.7), the remaining sum
will represent, as we have already remarked, not the expectation of the product (5.6), but
rather the product of three, or two expectations. And one of these will be a term of the series
E(xi pi) , E(xj pj) , E(xk pk)", , E(xr pr)
corresponding to the isolated subscript.
Our remark concerns all cases. If, however, the exponent of the selected subscript is unity,
the corresponding term of the series (6.1) is zero because
E(xi pi) = E(xj pj) = E(xk pk) = = E(xr pr) = 0.
And, of course, our product of the three expectations, equal to the sum of all the terms of
expression (5.7) excepting those excluded by us, also vanishes. It follows that in this case
expression (5.7) only consists of these eliminated terms which include as a multiplier at least
one of the coefficients of connection between the selected subscript and its adjacent
subscripts. Thus, our proposition is proved.
It will serve as a basis for our further conclusions about the sums composing (5.12) for the
special case in which some of the exponents (5.2) are unity. Concerning these sums we have
already indicated that they are reduced to consecutive summings over separate subscripts
arranged in a decreasing order. As to these summings, we have separated them in two groups
comparing the operations in the first one with a repetition of the same number, and the
summations in the second group with a decreasing geometric progression.
We have accordingly concluded that the ratio of each of our sums of (5.12) to the power of
n, equal to the number of the corresponding summations in the first group, must remain finite
. Of course, the number of exponents (5.2) can serve in all cases as the highest
boundary for the number of these summations. For our purpose, this boundary is sufficient if
only among (5.2) there are no exponents equal to unity. To specify: if under condition (5.3)
= = " = = = 2, then the number of exponents (5.2) is m/2 and we can establish that, as
lim{[E (xi pi)2(xj pj)2(xk pk)2 (xr pr)2/nm/2]
[Eciicjjcrr/ nm/2]} = 0
where i < j < < r < n,
cii = E(xi pi)2, cjj = E(xj pj)2 , , crr = E(xr pr)2.
If among the exponents (5.2) none are equal to unity, and, moreover, if some of them are
larger than 2, then their number is less than m/2 and, on the same grounds, as n
lim[E (xi pi) (xj pj) (xk pk)" (xr pr) /nm/2] = 0.
If, however, some of the exponents (5.2) are unity, we must establish another highest
boundary smaller than their number for the number of the summings in the first group.
In other words, we must now establish some lowest boundary differing from zero for the
number of summings in the second group. And, since for each term this number is equal to
the number of the included coefficients of connection, our problem is reduced to determining
the lowest boundary of this latter. In order to establish such a boundary, we shall consider the
totality of the coefficients of connection included in any of the non-vanishing terms of (5.12),
or, which is the same, in its corresponding terms in expression (5.7).
On the strength of the proven proposition, for each subscript whose exponent is unity,
there must be in this totality at least one coefficient of connection with its adjacent subscripts.
And one common coefficient can only correspond to two such subscripts if the series of
subscripts (5.4) arranged in an increasing order contains no intermediate subscripts, i.e., if
these two are next to each other. It is not difficult to conclude now, that for each term of
(5.12), not vanishing but really included into it, the number of summations of the second
group is not less than one half the number of all the exponents (5.2) equal to unity and can
only be equal to this magnitude when all the subscripts whose exponent is unity are
partitioned into separate pairs of adjacent subscripts, and the term selected by us contains all
the coefficients of connection of these pairs considered separately, but does not contain any
other coefficients.
If, therefore, the totality of the exponents (5.2) consists of unities,
twos, " , ,
numbers equal to 3, 4, respectively, then
m=
+"++ =
+ 2 + 3" + 4 +
and we may choose
( /2) + + " +
as the highest boundary of the number of summings in the first group for all the summings
considered; it is only attained under the indicated conditions.
On the other hand, it is not difficult to see that, if only not all the numbers " , , are
zeros, the sum (6.4) is less than half the sum + 2 + 3" + 4 + Therefore, we may
extend equality (6.3) onto all the totalities of the exponents (5.2) satisfying the condition
(5.3) and not solely consisting of twos and pairs of adjacent unities.
7. Having thus extended the equality (6.3) and issuing from it and formulas (5.1) and
(5.12), we immediately conclude that for odd values of m the ratio
E[(x1 p1) + (x2 p2) + + (xn pn)]m/nm/2
. Indeed, if the sum m of exponents (5.2) is odd, their totality
cannot only consist of pairs of adjacent unities and of twos. And, since the ratio (3.6),
remaining finite, cannot be arbitrarily small (3), the conclusion just mentioned might be
represented, as n
, by a simple formula, valid for any odd positive m
E 2( x1 p1 + x2 p2 + ... + xn pn )
=(1/
tmexp(t2)dt (7.1)
whose right side vanishes when m is odd.
To arrive at the same formula (7.1) for even powers of m as well, we ought to determine
definitively the main part of (5.11) for totalities of exponents (5.2) which only consist of
twos and of separate pairs of adjacent unities. According to 6, the main part of (5.12) for
such totalities of exponents (5.2) will be that, which includes all the coefficients of
connection of the separate pairs and does not include any other coefficients; the number of
summations in the first group in this term is indeed equal to m/2.
And, on the strength of the earlier reasoning on the severance of the chain into several
chains, it is not difficult to conclude that, in the cases considered by us, this term, which is
the main part of (5.11), is equal to the sum of the products adequately composed of the
expectations of the squares of some of the differences (xi pi), (xj pj), , (xr pr) and of
the expectations of the pairwise products of some other differences.
This conclusion might be represented, as n , by the following formula:
lim{E[ (xe pe)(xf pf)(xg pg) (xh ph) /nm/2]
[ cef cgh /nm/2]} = 0
cef = E(xe pe)(xf pf), cgh = E(xg pg)(xh ph),
and the subscripts
e, f, g, h,
whose number in each product
(xe pe)(xf pf)(xg pg) (xh ph)
is m, are only restricted by the inequalities
e f < g h < n.
The duality of the signs is eliminated when all the square multipliers of the products (7.4)
are ascertained.
Formula (7.2) incorporates the previous equality (6.2) as a particular case when e = f, g =
h, For the sake of convenience we shall additionally introduce new notation:
cii = E(xi pi)2 = Dii, 2cij = E(xi pi)(xj pj) = Dij
where i < j; the second subscript j in the expression for Dij will never be less than the first
one, i. Making use of this new notation, we can first of all represent the expectation of the
[(x1 p1) + (x2 p2) + + (xn pn)]2
as the sum Dij extended over all subscripts i and j obeying the inequalities
i j n.
Unlike it was in formula (7.2), the sign between i and j is not fixed but remains dual.
Then, on the grounds of formulas (5.1), (6.3) and (7.2) it is not difficult to establish the
following formula {as n } for even values of m
lim{E[(x1 p1) + (x2 p2) + + (xn pn)]m/nm/2]
[m!/(2n) m/2] Def Dgh } = 0
where all the values satisfying the inequalities (7.5) should be assigned to the subscripts (7.3)
whose number in each product Def Dgh is m. Once more, the signs between e and f, g
and h, are not fixed but remain dual as in the just mentioned sum Dij.
It remains to compare the sum
Def Dgh = S
that appears in formula (7.6) with the power ( Dij)m/2. This power contains our sum S with a
coefficient (m/2)! as well as other products of the same kind,
Def Dgh
which do not, however, obey our inequalities (7.5).
We ought to prove that the ratio of the sum of these products, taken of course with the
coefficients adequate for composing the difference
( Dij)m/2 (m/2)!S,
to n m/2 tends to zero as n
. Considering the totality of the products (7.8) making up the
difference (7.9), we shall establish a certain order among the factors so as to avoid their
multiple repetition and to ascertain the composition of the studied totality. We shall always
arrange the first subscripts e, g, in an increasing order; and, if they coincide, we shall
arrange the second subscripts in the same way.
Having thus established the order of the multipliers of the product (7.8), we can easily
solve the question of whether this product is included in the sum S, or, on the contrary,
whether it belongs to that totality which makes up the difference (7.9) and which we ought to
study. If the product (7.8) belongs to the totality under study, then one of the second
subscripts f, h, must be not less than the subsequent first subscript; for example, f g.
We shall make use of this remark for composing the sum which includes all the terms of
the difference (7.9) and which therefore provides, after replacing the magnitudes Dij by their
absolute values, |Dij|, a number certainly larger than this difference. Consider the totality of
those products (7.8), which include one and the same multiplier Drs, such that the second
subscript of the next factor is not larger than s. It is not difficult to note that this totality
constitutes a part of the one that follows after fulfilling the operations indicated in the
(m/2)[(m/2) 1]Drs ( Dij)m/2 2 (Drr + Dr r+1 + Dr r+2 + + Dr+1 r+1 +
Dr+1 r+2 + + Dss + Ds s+1 + Ds s+2 + ).
Its first factor, (m/2)[(m/2) 1], determines the coefficient with which our products might
enter in the first term of (7.9). Replacing in this expression all the Dij by their absolute values,
and summing the obtained expressions over all possible subscripts r and s, we arrive at the
sum which we shall denote by
and which certainly exceeds the absolute value of the
difference (7.9). On the other hand (3), we have inequalities
|Drs| < sr/2, |Dij| < n (1+ /[4(1 )],
|Drr| + |Dr r+1| + |Dr r+2| + < 1/[2(1 )],
|Dss| + |Ds s+1| + |Ds s+2| + < 1/[2(1 )]
n(1 + )
< (m/2)[(m/2) 1]
4(1 ) 4(1 )
(m/2)[(m/2) 1]
4(1 )3
4(1 )
m / 2 2
(1 + 2 + 3
+ ) <
Consequently, the absolute value of the difference (7.9) must also be less than the last
expression just above and we can therefore establish the formula
lim{[( Dij) m/2/n m/2] [(m/2)!S/n m/2]} = 0 {as n }
where S is defined by (7.7).
It remains to compare formulas (7.10) and (7.6) and to take into account the well-known
t mexp(t2) dt = [1 3 5 (m 1)/2m/2]
to obtain immediately, for any even m, the formula (7.1) which we have previously
established for odd exponents m. Having ascertained formula (7.1) for all integral exponents
m, we may issue from it and conclude, as it is known, that, for any given numbers t1 and t2 >
t1 and n
, the probability of the persistence of the inequalities
t1 <
E 2( x1 p1 + x2 p2 + ... + xn pn )2
< t2 ,
or, which is the same, of
m p1 p2 ... pn
E 2( m p1 p2 ... pn )2
as n 0 1 tends to
exp(t2) dt.
As to the expectation of the square
[m (p1 + p2 + + pn)]2,
it is determined by the previously established formula (3.4).
We are thus extending the theorem about representing the limit of probability by the
Laplace integral onto the general case of trials connected into a chain under some restrictions
that are indicated above and occasioned by the course of our reasoning.
8. In concluding our article, we return to the expectation of (7.11), which, in accord with
formula (3.4), is equal to
p1 q1 (1 + 2
4 n) + + pn-1 qn-1 (1 + 2 n) + pn qn.
We shall try to ascertain that the condition put forth in 3 does not introduce any new
restrictions except those indicated in the introduction. For this purpose we deduce from
formula (1.2) simple equalities
pk+1 qk+1 = (p2k+1 + k+1 pk) (q3k+1 + k+1 qk) =
k+1 pk qk + k+1(pk q3k+1 + qk p2k+1) + p2k+1 q3k+1
pk+1 qk+1 = (p3k+1 k+1 qk) (q2k+1 k+1 pk) =
k+1 pk qk k+1 (pk p3k+1 + qk q2k+1) + p3k+1 q2k+1.
Now we infer that the difference [pk+1 qk+1 2k+1 pk qk] is not less than one of the products
p2k+1 q3k+1 and p3k+1 q2k+1 and cannot therefore be arbitrarily small, if only, as stated in the
introduction, among the numbers
p3k+1, p2k+1, q3k+1, q2k+1
not a single one is arbitrarily small. Taking this into account, we denote
pk+1 qk+1
pk qk =
and replace
consecutively, in the left side of equality (3.4),
pn qn, by an equal magnitude 2n pn1 + n1,
pn1 qn1, by an equal magnitude 2n1 pn2 qn2 +
p2 q2, by an equal magnitude 22 p1 q1+ 1.
We thus transform formula (3.4) into
E[m (p1 + p2 + + pn)]2 = Tn2
+ Tn12
+ + T22
+ T12
0 = p1q1,
Tn= 1, Tn1 = 1 +
=1+
nTn,
Tn2 = 1 +
n1 n
n1 Tn1
and in general
Tk1 = 1 +
kT k
k+
k k+1 +
We also have
T2k+1 + T2k = (1 + k2)T2k + 2 kTk + 1 =
(1 + k2){Tk + [ k/(1 + k2)]}2 + [1/(1 +
)],
from which we conclude that for any subscript k the sum in the left side cannot be arbitrarily
small, but must always remain not less than [1/(1 + 2)]. Therefore, if only the conditions
indicated in the introduction are fulfilled, the ratio
E[m (p1 + p2 + + pn)]2/n
cannot be arbitrarily small either.
Thus, the conclusions of this article do not demand other restrictions except those
indicated in the introduction. They consist in that the numbers (8.1), at all values of k, remain
larger than some constant positive number. The question about the possible relaxation of the
restrictions imposed by us remains open.
Notes and Comment by N. A. Sapogov [8, pp. 666 668]
1. We think that this term can describe the chains that we have considered in contrast to
chains of other dependent magnitudes.
2. We shall remove this doubt at the end of our paper.
Markov proves by the method of moments that the CLT is applicable to the sums (X1 + X2
+ + + Xn) of random variables Xi connected into a simple non-homogeneous chain under
the assumption that the possible values of Xi are 0 and 1 and
0 < p0 < p3i < 1 p0, 0 < p0 < p2i < 1 p0
where p3i and p2i are the transition probabilities
p3i = P(Xi = 1 / Xi1 = 1), p2i = P(Xi = 1 / Xi1 = 0)
and po is a constant number not depending on n.
Markov left open the question of whether the CLT was applicable to the sums of Xi if po
can approach zero infinitely. Bernstein [1] arrived at the same result by another method and
proved that the CLT persisted if po = 1/n with the constant < 1/5, but that it was perhaps
non-applicable for = 1/3. In two other writings he [2; 3] proved the same proposition for
the case of < 1/3.
I [9] have shown the applicability of the CLT under the condition that po = (n)/n1/3 where
(n) was an arbitrary function infinitely increasing as n
. Linnik [4] considered a more
general case of a non-homogeneous chain of restricted variables Xi with an arbitrary finite
number of possible values xi(k),
1 k ki for each Xi and proved that the CLT is applicable if all the transition probabilities
differ from 0 and 1 not less than by 1/n with the constant < 1/3 and
(1/ki)
[xi (1/ki)
xi(k)]2 c > 0.
I [10] obtained a similar result also for the case of chains of h-dimensional vectors.
1. Bernstein, S.N. (1926), Sur lextension du thorme limite du calcul des probabilits
aux sommes de quantits dpendantes. Math. Annalen, Bd. 97, pp. 1 59.
2. Bernstein, S.N. (1926), Sur les sommes de quantits dpendantes. Izvestia Akad. Nauk
SSSR, ser. 6, vol. 20, pp. 1459 1478.
3. Bernstein, S.N. (1928), Addition larticle [to the previous paper]. Doklady Akad.
Nauk SSSR, ser. A, pp. 55 60.
4. Linnik, Yu.V. (1949), Izvestia Akad. Nauk SSSR, ser. math., vol. 13, pp. 65 94.
5. Markov, A.A. (1906), The extension of the law of large numbers, etc. Translated in this
6. Markov, A.A. (1907, in Russian), Recherches sur un cas remarquable dpreuves
dpendantes. Acta math., t. 33, 1910, pp. 87 104.
7. Markov, A.A. (1908), The extension of the limit theorem, etc. Translated in this book.
8. Markov, A.A. (1951),
9. Sapogov, N.A. (1947), Doklady Akad. Nauk SSSR, No. 2, vol. 58, pp. 193 196.
10. Sapogov, N.A. (1949), Ibidem, No. 2, vol. 69, pp. 133 135.
14. A.A. Markov On the Coefficient of Dispersion
As Markov indicated in the beginning of his paper, the first to consider the expectation of
the coefficient of dispersion was Chuprov. Indeed, in 1916 Chuprov informed Markov in a
private letter that he had proved that EQ = 1. Markov, however, was the first to publish this
result and in a few months he communicated Chuprovs relevant contribution to a periodical of
the Petersburg Academy of Sciences, see pp. 112 113 of my book Chuprov. Life, Work,
Correspondence. Gttingen, 1996. Chuprovs paper appeared in my translation in Chuprov,
Statistical Papers and Memorial Publications. Berlin, 2004, pp. 39 47.
[1] Without dwelling on the importance of the coefficient of dispersion for statistics, I aim
here, first, to prove the proposition stated by Professor Chuprov, that its expectation for
independent trials with a constant probability is exactly equal to unity provided that we do not
extract the square root {of the coefficient} but adhere to the definition adopted in my book [3];
and, second, to establish, for the case of an equal number of trials {in each series}, a rather
simple expression for the expectation of the square of the deviation of this coefficient from
unity. The expectation thus obtained is somewhat in excess of its exact value 1.
I connect the proposition about the expectation of the coefficient of dispersion with the name
of Professor Chuprov because, as far as I know, he was the first to consider not the numerator
and the denominator of this fractional expression taken separately one from another, but the
fraction itself, and arrived at the abovementioned conclusion, at least for the case of equal
series. As to the second problem, Professor L. Bortkiewicz had long ago discovered its solution
of sorts, but it was connected with such assumptions, which we, taking care of rigor and clarity
of definitions, and of absolute rigor of deductions, cannot admit.
[2] Let us consider the general case of several series of independent trials with a constant
probability. Denote the series, in number, by 1, 2, , ; the number of trials in series i by si;
the corresponding number of appearances of event E by xi with
n = s1+ s2+ + s and m = x1 + x2 + + x
and, finally, the constant probability of event E by p, and let the coefficient of dispersion for
the considered totality of series be
Q=
Q2 =
n ( n 1)
si [( xi / si ) ( m / n )]2
( 1)m( n m)
( nxi / si ) m2 ]
( n 1)[
( n 1)2
( 1)2
[n 2 xi / si ] + 2
[n 2 xi x j / si s j ] 2m2
m 2 ( n m) 2
[nxi / si ] + m4
These expressions lose their meaning if m = 0, or if m = n, when both their numerators and
denominators vanish. In these exceptional cases we shall assume that Q = Q2 = 1. To calculate
the expectations of Q and Q2 we must multiply their expressions by the probability P of the
totality of numbers
x1, x2, , x
and compile the sums PQ and PQ2 for all possible totalities (1). The probability P is equal
P=
s1!
s2 !
s !
pmqnm
x1!( s1 x1 )! x2!( s2 x2 )!
x !( s x )!
where q = 1 p.
We shall break up the summation of PQ and PQ2, which should not be confused with the
(as in the formulas above), in two stages. For the first summation we
i, j
shall suppose that m is invariable, and we shall only vary this number alone during the second
one. This sequence of operations can be written down as
PQ and
PQ2 with m = 0, 1, 2, , n for the external sums and
x1 + x2 + + x = m for the internal ones.
During the first summation, the denominators m (n m) and m2 (n m)2 of the expressions Q
and Q2 respectively remain constant, so that the problem of determining the inner sums reduces
to calculating
P, Pxi2, Pxi4, Pxi2xj2
where i and j denote some two indices taken from our system 1, 2, , and remaining
invariable during the summation.
To determine all these sums we introduce ( + 1) arbitrary magnitudes 1, 2, , , t and
construct their function
W = [pt exp( 1) + q] s1 [pt exp( i)+ q] si [pt exp( ) + q] s
which can be decomposed into terms
Pt x1 + x 2 +... + x e x11 + x 2 2 +...+ x .
To determine the first of the sums in (2) by means of W, it is only necessary to equate all the
s to zero; develop the thus obtained expression W = (pt + q)n in powers of t; and take the
coefficient of t m, equal to Cn p mqnm.
In a similar way, the coefficients of t m in the expressions
2W / i , 4W / i , 4W / i j at
will be equal to the other sums in (2) respectively. We must only consider these sums for
0<m<n
since, in accord with the definition adopted,
PQ = PQ2 = qn if m = 0 and pn if m = n.
To calculate the expectation of Q, we begin with the first of these sums. We have, for
= = = 0,
2W / i = si pt(pt + q) + si(si 1) p t (pt + q)
so that, for x1 + x2 + + x = m,
P(nxi2/si) = [ Cn 1
+(si 1) Cn 2
]npmqnm.
P[(nx12/s1) + (nx22/s2) + + (nx 2/s ) m2] =
( n 2)! n
[ (n 1) + (n )(m 1) m(n 1)]pmqnm =
( m 1)!( n m)!
n( 1) Cn 2
PQ = Cn pmqnm.
We have deduced this formula under condition (3) and it was previously established for m =
0 and m = n. Carrying out the second summing, we obtain Professor Chuprovs proposition
PQ = (p + q)n = 1.
Turning now to the expectation of (Q 1)2, that, on the strength of the proven is equal to the
Cn pmqnm,
we restrict our attention to the case in which all the numbers s1, s2, , s take one and the same
value s. In this particular case, as shown by direct calculation 2, Q is reduced to unity when m =
1 and m = n 1. Therefore, not only the highest powers of p and q, but also the products pn1q
and p qn1 vanish in the expression EQ2 (p + q)n. And, because of the formulas (where
= 0)
1= 2==
(1/s) 4W / i = pt(pt + q)n1 + 7(s 1)p2t2(pt + q)n2 +
6(s 1)(s 2) p3t3(pt + q)n3+ (s 1)(s 2)(s 3)p4t4(pt + q)n4
(1/s2) 4W / i j =
p2t2(pt + q)n2 + 2p3t3(s 1)(pt + q)n3 + (s 1)2p4t4(pt + q)n4,
we find without much difficulties that, for x1 + x2 + + x = m,
P(Q2 1) =
( n 4)!( n 1)np mq n m R ( m )
( 1) 2 ( m 1)! m2 ( n m)!( n m) 2
R(m) = 2(n 3)(n 2)(n 1)2 + 7 2(s 1)(n 3)(n 2)(n 1)(m 1) +
6 2(s 1)(s 2)(n 3)(n 1)(m 1)(m 2) +
(s 1)(s 2)(s 3)(n 1)(m 1)(m 2)(m 3) +
n ( 1)(n 3)(n 2)(n 1)(m 1) +
2n ( 1)(s 1)(n 3)(n 1)(m 1)(m 2) +
n ( 1)(s 1)2(n 1)(m 1)(m 2)(m 3)
2m2 (n 3)(n 2)(n 1)2 2m2(n )(n 3)(n 2)(n 1)(m 1) +
m3(n 3)(n 2)(n 1)2 ( 1)2m(n m)2(n 3)(n 2).
To facilitate the study of the rather involved expression for R(m) it is possible to take
advantage of the circumstance that Q does not change its value when all the xis are replaced
by the differences(s xi) so that m is replaced by (n m). Accordingly, we have
pnmqm
P(Q2 1) = pmqnm
P(Q2 1)
where x1 + x2 + + x is equal to m and (n m) respectively, and therefore
mR(m) = (n m)R(nm).
It follows, first of all, that R(m) includes the factor (n m). Then, establishing by direct
calculation that R(1) = 0, we reveal the presence of not only the factor (m 1) but also of the
factor (n m 1) in the expression R(m) which is an integral function of the third degree with
respect to m. Therefore, R(m) is divisible by the product (n m)(n m 1)(m 1), and for its
complete determination it only remains to consider the coefficient of m3 which is equal to the
A = 2(s 1)(s 2)(s 3)(n 1) + n ( 1)(s 1)2(n 1) +
(n 3)(n 2)(n 1) 2 ( 1) 2 (n 3)(n 2) 2(n )(n 3)(n 2)(n 1)
where n = s . Combining the three last terms of A, we find that their sum is equal to
(s 1)2(n 3)(n 2)
which enables us to isolate the factor 2(s 1) from A.
We carry out the subsequent calculation in the following way:
A = 2(s 1)[(s 2)(s 3)(n 1) +
(n s)(s 1)(n 1) (s 1)(n 2)(n 3)] =
(s 1)[(s 2)(s 3)(n 1) (s 2)(s 1)(n 1) +
(n 2)(s 1)(n 1) (s 1)(n 2) (n 3)] =
2 2(s 1)[ (n 1)(s 2) + (n 2)(s 1)] =
2 2(s 1)(n s) = 2 2s(s 1)( 1).
Thus, for x1 + x2 + + x = m,
( m 1)( n m 1) m m nm
2 2 s( s 1)
Cn p q
(s 2)(s 3)( 1)
m( n m)
E(Q 1) =
from which, since the expression (m 1)(n m 1)/[m(n m)] invariably remains less than
unity, immediately follows the inequality
E(Q 1)2 <
5 it is not difficult to derive now a very simple inequality 3
E(Q 1)2 > 2/(
For large values of n this highest boundary for the expectation of (Q 1)2 also represents its
approximate value since the main terms in the sum included in (4) are those for whom m is
close to np and the difference (n m) is close to nq, and for whom therefore the expression (m
1)(n m 1)/[m(n m)] differs but little from unity if (n s ) is a large number 4.
[3] Being prompted by an article of Tikhomirov [3], I take this opportunity to say a few
words about the fashionable theory of correlation. Its positive side is not significant enough
and consists in a simple usage of the MLSq to discover linear dependences. However, not
being satisfied with approximately determining various coefficients, the theory also indicates
their probable errors, and enters here in the realm of imagination, hypnotism and belief in
mathematical formulas that actually have no sound scientific foundation. Of such nature is, for
example, the Pearson formula which plays an important part in Tikhomirovs paper and which
is adduced after a vain reference to my book [5] 5.
The hypnotism of the theory of correlation manifests itself in the following words of the
same paper:
When r is equal to zero, it is said that there is no correlation between the elements, and in this
case it is absolutely impossible to estimate the variability of the other element when issuing
from the deviations of the first one.
Actually, however, it is not difficult to construct any number of connections which will not
at all be detected by the coefficient of correlation, and such, that, nevertheless, the changes of
one element will determine those of the other one. Some indication of such cases is also found
in Tikhomirovs article (p. 34). The passage even begins there by declaring that It is important
to point out , but, contradicting these words, remains futile and ends by stating that
It is said in this case that the variables are not correlated, but, at the same time, are not
independent one from another.
The following opinion (p. 26) should also be attributed to the realm of belief:
The equation of regression only means that, knowing the value of xi, it is possible to say that
the most probable value of yi will be r r/ 1.
The example given in Table 4 (p. 43) can serve for describing propositions based on the
theory of correlation. For 23 pairs of magnitudes x and y entered there, the coefficient of
correlation occurred to be small (0.09), whereas its probable error, comparatively large (0.14).
The author inferred that the existence of correlation between these magnitudes cannot be
considered proven; I shall not by any means try to refute such an indefinite conclusion.
However, if the last ten pairs instead of all the 23 of them are taken, the coefficient will
exceed 0.7, whereas the Pearsonian probable error will fall down to 0.1 which is less than 1/6
of this coefficient. Then, in accord with the rule indicated on p. 24, we shall have to make an
absolutely different conclusion:
In practice, it is usually considered that the existence of correlation between the elements
studied is entirely certain when the coefficient of correlation is not less than six times its
probable error. This is equivalent to having many thousand chances to one in favor of the
connection actually existing.
To avoid misunderstanding, or possible debates about 10 being a small number and 23 being
a sufficiently large number, I remark that even in the theory of observational errors I do not
attach any great importance to the so-called probable errors and only consider them as a means
for conjecturally comparing the merit of different observations. As to the coefficient of
correlation, it has a quite definite value and can be calculated without any error when studying
a given totality of numbers. If, however, this totality is considered as a part of an absolutely
unknown totality, then the coefficient of correlation for the latter cannot be determined either
when the former consists of 10 pairs, or of 23, or of a much larger number of pairs. If desired,
it is of course possible to consider the number calculated for the given totality as an
approximate value of this coefficient whose very existence is doubtful; but one cannot help to
call the probable error of such a determination, and the connected calculation of chances, pure
fiction 6.
Comments by Yu.V. Linnik [3, pp. 668 670]
1. The coefficient of dispersion in the form derived by Markov enables to establish some test
of homogeneity of the series of independent trials; namely, the test for examining the
hypothesis on the constancy of the probability of the event observed in these series. If this
hypothesis is true, the values of the expectation and the variance of Q will be those indicated
by Markov and allowing, for example, the use of the Chebyshev inequality.
2. The calculation can be carried out thus. When m = 1, and si = s, i = 1, 2, , we have n =
s, x1 + + x = 1 so that xi = 1 for one i, and all the other xs are zero. Therefore, Q is
necessarily
n ( n 1) s
{[(1/s) (1/n)]2 + [( 1)/n2]} =
( 1) ( n 1)
{[(1/s) (1/ s)]2 + [( 1)/
s ]} = 1.
When m = n 1, xi = s 1 for one i, and all the other xs are s. In this case
( 1)[1
s 1 2 s 1 s 1 2
] +[
] =
s 2 1 ( 1) 2
1 (s )2
(s ) 2
3. We have
2 [1 (1 / s )]
( 1)[1 ( 2 / s )][1 (3 / s )]
It remains to prove that for
1 (1/s) < [1 (2/ s)] [1 (3/ s)].
But, under this condition,
[1 (2/ s)] [1 (3/ s)] [1 (2/5s)] [1 (3/5s)] =
1 (1/s) + (6/25) (1/s2) > 1 (1/s).
4. This proposition follows, for example, from the Chebyshev inequality written down in the
P[(m np) > n ] < pq/n
with an appropriate choice of .
5. Tikhomirov (p. 23) writes down the Pearson formula as the probable error is equal to
0.674(1 r2)/ n. Strictly speaking, this formula is meaningless already because the sample
coefficient of correlation r is a random variable. The rigorous sense, and the condition for
applying the Pearson formulas are as follows: For samples of size n
from a normal
bivariate general population (X; Y) with (| | ! 1) being the coefficient of correlation between
X and Y, the sample coefficient of correlation r tends to normality, its expectation approaches
, and its variance is asymptotically equal to (1 2)2/n, see [1]. In the absence of grounds for
assuming that the general population is normal, the Pearson formula of course has no scientific
6. Markov indicates here that it is impossible to speak about the distribution of the sample
coefficient of correlation without knowing the type of the initial general distribution (X; Y).
In our time, there exist calculations of such a distribution for a normal (X; Y). It occurs that the
distribution of the sample coefficient of correlation only depends on the general coefficient
of correlation and the size of the sample n. In this case, given a large number of observations,
it is possible to estimate .
1. Kuzmin, P.O. (1939), Sur la loi de distribution du coefficient de corrlation dans les
tirages dun ensemble normal. Doklady AN SU, vol. 22, No. 6, pp. 298 301.
(Calculus of Probability). Psb. Later
editions: 1908, 1913, 1924; German translation, 1912.
4. Tikhomirov, G. (1915), The method of correlation and its application in meteorology.
Geofizich. Sbornik, vol. 2, No. 3, pp. 21 48. (R)
15. A.A. Markov. On the Coefficient of Dispersion for Small Numbers
Strakhovoe Obozrenie, No. 2, 1916, pp. 55 59
For small numbers, the coefficient of dispersion cannot be large. Since some statisticians [1;
2, pp. 131 134] are apparently inclined to exaggerate the importance of this fact perceiving
there a peculiar law of small numbers, and a proof of a special stability of these numbers, it
seems that it would be helpful to show its independence of the theory of probability. This is
indeed the aim of my note.
We begin with a simplest assumption that, in M series of N observations in each, some event
occurred a times in m series and had not occurred at all in (M m) series. The square 1 of the
coefficient of dispersion will here be
( M m) ( ma )2 + m[( M m)a]2
Q =
( M 1) ma [ M ( ma / N )]
Disregarding the fraction ma/N because of the supposed very large value of N, we have
Q2 = [(M m)/(M 1)]a.
Consequently, in this simplest example the coefficient of dispersion is, generally, somewhat
less than a and only reaches this extreme value at m = 1. Then, it is not difficult to see that the
same upper boundary a persists for Q in case of differing numbers {of successes} not
exceeding a. The coefficient Q can only reach its indicated maximal value when the system {of
successes} consists of zeros and a single a (m = 1).
If, however, the system several times includes a as well as other numbers less than a but
differing from zero, the coefficient of dispersion must be much less than a. Thus, for a system
consisting of zeros and unities, the coefficient will be without fail less than unity. The same
holds if a small number of twos is added to a large number of unities. For example, the
coefficient of dispersion for a system consisting of 15 zeros (non-occurrences), 9 unities and
two successes in one single case [1, pp. 18 19] is naturally less than unity.
When changing this example but leaving the previous number of zeros and the total number
of unities and twos (10 unities; 9 unities and two successes in 1 case; ; 10 twos; with 15
zeros in each system), we will have 11 systems with the maximal value of the coefficient of
dispersion 2 15 / 24 = 1.25 = 1.11 for the system consisting of 15 zeros and 10 twos.
Some light can also be thrown by an examination of all possible combinations from four
numbers not exceeding 3. If the order of the numbers is taken into account, but the totality of
four zeros be eliminated, there will be 44 1 = 225 such combinations.
For the 15 combinations in which neither number exceeds 1, the coefficient of dispersion
naturally does not exceed unity and only reaches it in four cases,
0001, 0010, 0100, 1000.
For the 81 16 = 65 combinations in which a two is included but a three is not, the
coefficient remains not larger than 2. It reaches this limit in four cases, in combinations of
three zeros and a two; and it takes the value 4 / 3 in six cases (two twos and two zeros), and
the value 11 / 9 in 12 cases (2 zeros, a unity and a two). The other 65 22 = 43 combinations
in which a two occurs but not a three, the coefficient of dispersion is less than 1. For 256 81
= 175 combinations with a three, the coefficient remains not larger than 3 and only reaches
this limit in four combinations (three zeros and a three). Then, the coefficient reaches
2, in 6 cases, 0033, 0303, ...; and in 12 cases, 0013, 0103, ;
9 / 5 , in 12 cases, 0023, 0203, ; 9 / 7 , in 12 cases, 0133, 1033, ...;
19 / 15 , in 12 cases, 0113, 1013, ; 10 / 9 , in 24 cases, 0123, 1023, ...;
The other 175 82 = 93 combinations of the considered type provide a coefficient of
dispersion not exceeding unity. A similar study of a large number of terms and their
considerable upper boundary is of course very difficult. Therefore, instead of accomplishing it
on a large scale, I shall dwell on the combinations resulting from another example also
adduced in [1] as a confirmation of the law of small numbers:
144 times, 0; 91 times, 1; 32 times, 2; 11 times, 3; 2 times, 4.
I shall consider all the 124 pertinent combinations without changing the numbers of zeros, of
threes and fours as well as leaving intact the previous total number of unities and twos. In
general, these combinations are
144 times, 0; x times 1; (123 x) times, 2; 11 times, 3, 2 times, 4.
For whichever x the coefficient of dispersion will be
623 3x 287 x
For investigating the behavior of this function at 124 considered values of x we calculate its
derivation with respect to x; it is equal to
280 ( 287 x )2
For x 28 the derivative is positive and it becomes negative for x 29. Therefore, the
coefficient of dispersion reaches its maximal values at x = 28 and 29 and its minimal values at
x = 0 and 123. Calculations provide the following values for the square of the coefficient:
at x = 0, 28, 29, 123, Q2 = 1.15, 1.16, 1.16, 0.97, respectively.
We see that in all 124 cases the coefficient remains close to unity and less than 1.09.
It is thus confirmed that the coefficient of dispersion can be close to unity for most various
combinations which have no relation to the theory of probability and that the smallness of the
coefficient for small numbers does not represent anything special 2.
As to the application of the remarkable Poisson formula to small numbers, it might be
considered well-founded provided that a large totality of homogeneous series of observations is
investigated; otherwise, it is only a skilfull play with numbers.
1. I keep to the generally adopted term although I consider the taking of the root
unnecessary.
2.{Markov repeated this statement in several letters to Chuprov and in a letter to Kaufman,
see Sheynin (1990, p. 111). Quine & Seneta (1987) argued, contrary to Markov and without
citing him, that the smallness of the coefficient of dispersion is understandable from a
stochastic point of view, and I am inclined to agree with them.}
1. Bortkiewicz, L. von (1898), Das Gesetz der kleinen Zahlen. Leipzig.
2. Kaufman, A.A. (1916),
(Theory and Methods of
Statistics). M. Third edition.
3. Quine, M.P., Seneta, E. (1987), Bortkiewicz data and the law of small numbers. Intern.
Stat. Rev., vol. 55, pp. 173 181.
4. Sheynin, O. (1990, in Russian), Chuprov. Gttingen, 1996.
16. The Correspondence between A.A. Markov and B.M. Koialovich
Professor Boris Mikhailovich Koialovsky (1867 1941) was Markovs gifted student [1, p.
220] and an eminent scientist [2, p. 415]. He taught at several academic institutions, notably at
Petersburg Technological Institute, and achieved certain success in studying differential
equations [3]. His correspondence with Markov characterizes himself, to some extent describes
Markovs exposition [4] of the MLSq and once more confirms that Markov was diligently
studying the relevant literature. The letters are kept at the Archive, Russian Academy of
Sciences, Fond 173, Inventory 1; the additional numbers provided below show the place of the
letters in that Fond.
Letter 1, Koialovich Markov, 25.9.1893 (10, No. 1)
Dear Andrei Andreevich! I received your letter and am thankful for the attention to my
lectures. I personally considered them as an urgent work only written for giving the students
some manual for recalling my lectures. []
I regret very much that my lectures made such an impression on you; and I regret even more
that I felt in your letter an extremely unexpected opinion about me with which I cannot at all
agree because I consider it undeserved and unjust. [] Do you, Andrei Andreevich, really
think that I am unacquainted with the true essence of the {Stirling} formula? []
It is of course not for me to judge to what extent I was able to explain the notion of
probability. I shall only dwell on that example to which you object. {Taken by itself, the
authors explanation is not sufficient; apparently, however, the matter was not important.}
There remained for me a choice between two methods: either to base my reasoning on the
notion of probability of error, or on considering, as Chebyshev did in his lectures, whose
manuscript notes I had in my possession, a sum of randomly selected magnitudes. The second
possibility, which I naturally prefer, was impossible for me owing to lack of time. Then, I add
that in any explication of the MLSq we have to rely on more or less arbitrary assumptions.
This, as I think, cannot and should not be avoided since the matter concerns a practical method
for treating observations rather than a construction of an abstract mathematical theory. I am far
from desiring to prove somehow the MLSq and I have therefore used the term justify (p. 55)
rather than prove. I think that justifying and proving are absolutely different notions.
I also note that from among the hypotheses which we undoubtedly have to introduce when
explicating the MLSq, the one concerning the properties of the probability of error is in any
case the most natural although I know very well the difficulties connected with it 1. []
I called the results provided by the MLSq quite trustworthy, because, as far as I know, such
are all the main numerical results of our natural sciences, physics and astronomy.
You say that, once the data are sound, the results will always be good even without the
MLSq. I agree absolutely, but from among these good results some might nevertheless be
better than the others depending on how we combine the observations. []
I did not at all attempt to connect the works of Sleshinsky [5] and Yarochenko [6; 7] with
those of Chebyshev 2. For me, they have only one common point: they explicate the
MLSq.unlike Gauss, and it was in this sense that I have cited them.
I extremely regret that your works remained unknown to me and I apologize for omitting to
cite them. They were unknown to me because I have not seen any references to them, and, as
far as I can remember, you never mentioned them in our talks 3.
One of Chebyshevs works, I do not remember which one, had been in my hands, but for
such a short time that I was unable to acquaint myself in detail with it. For me, it was sufficient
to see that he explicated the MLSq not as Gauss did.
Concerning the function which I denoted by ( ), I allow myself to turn your attention to
the following. Let us imagine a series of magnitudes
n , , 2 , , 0, , 2 , , ,
+ , ,
+n
and assume at first that is a finite although a very small magnitude. If we suppose that the
error of observation can only take values indicated in the series (1), then the notion about its
probability is quite clear for any . If we will now decrease indefinitely, then our hypothesis
will come arbitrarily close to an assumption of a continuous error and will become the
differential of the magnitude . The function that I denoted by ( ) will indeed express the
probability {as studied} in the theory of probability. The aim of the theory is not to create the
notion of probability anew, but to explain it and to make it accessible to measurement. By
stating that a well-known ratio is probability, we do not explain the difficulty, but only keep
ourselves away from such an explanation and I did not consider it possible to do so. []
Concerning the notion of dependent and independent trials, I quite agree with you that I have
left a gap here, although the explication below made it clear which trials I had discussed. []
I am turning now to the main point of our disagreement, to the MLSq. You reproach me as
your student for avoiding your adopted description of the method. Believe me, Andrei
Andreevich, from among your students there are hardly many who respect you more than I do;
however, even for all this respect, I reserve for myself the right to a free choice in the matters
of science and I can never renounce this right. You yourself would hardly be pleased if your
students were to restrict their teaching by blindly imitating your lectures. And the reasons why
I have chosen a different explication of the MLSq are these.
1) I believe that your exposition will be too difficult for my listeners 4.
2) There is one point in your account which I was never able to understand either myself or
during our conversations. Here it is. On p. 159 of your lectures (1891) [8] you say:
Let us introduce, in accord with the above, magnitudes x , x , , x(n) representing the possible
result of the first, the second, , the n-th observation.
It is this place that I was unable to understand. What do you mean by the possible results of
observation? Under what conditions are they possible and how do they differ one from
another? I could not understand this without introducing once more the concept of the
probability of error and I therefore preferred to decide in favor of an error equal to and to
assume that = d is an infinitesimal.
The function ( ) is thus not only not senseless, but it is even very beneficial in that it
eliminated the need for introducing, while explicating the MLSq, the concept of density 5. []
I hope that my explanations will serve to eliminate the misunderstandings and disagreements
which appeared between us.
Letter 2, Koialovich Markov, 2.10.1893 (10, No. 8)
[] I know very well that my course is corrupted by many shortcomings which should be
eliminated later on. [] {Concerning the example discussed in Letter 1}I say clearly enough
that the point only concerns the choice of the most probable event; and an event remains most
probable no matter how low is its probability if only it is higher than the probabilities of the
other events. []
You write that I attempt to obscure both this notion (of equally possible cases) and the
concept of probability. I believe that I have the right to protest against such a charge because it
rings oddly with respect to a man who nevertheless tries his best to reveal the truth. I can
obscure the truth unintentionally but I cannot attempt to do so; that would imply such motives
which you are unable to assume that I have. []
I am asking you to turn your attention to the following difficulties, which, as it seems to me,
naturally arise when keeping to your viewpoint. You write: Why may we not say in the theory
that the well-known relation is probability? I answer: We certainly may, and we may say that
anything is probability, but for whom will such a theory be compulsory and interesting? Will
not everyone have the right to say: Am I concerned with theories operating on notions
concocted by you yourself which perhaps have no object {no representation} in reality? Will
not all this theory become bungen fr den Verfasser (as apparently Weierstrass expressed it) 6
Therefore, you will probably agree that I have grounds for remarking that we do not explain
the difficulties but only keep ourselves away from them. []
I shall now dwell on the MLSq. I am naturally acquainted with Laplaces account and
completely agree with you that I should have indicated it. I had certain motives for citing
Sleshinsky and Yarochenko; I myself have a low opinion about them, and especially with
regard to the former, but these motives are subjective and not obligatory for anyone else.
You write that you perceive that I am not at all acquainted with the works of Chebyshev and
others. This surprises me very much. Indeed, I held them in my hands and read them. Or, do
not you trust me?
Concerning your account of the MLSq I allow myself to indicate the following. I know of
course that your p. 159 carries a reference to previous explanations (probably to p. 157); I have
not mentioned this because, to my utmost understanding, nothing is explained either on p. 157
or elsewhere. Even your latest explanation, for which I am of course sincerely thankful, did not
explain to me my most serious perplexity, namely:
As far as I understand you, you consider each separate observation as a value of a possible
result. Thus, a series of results
a1, a2,
is possible for each measurement, and one of them is realized. I am prepared to understand all
this concerning one observation. However, if there are, for example, two observations, then I
cannot understand the difference between the series of all the possible results of the first
observation (A) and the similar series for the second measurement (B) 7. The problem will be
certainly solved at once if you say that the probabilities of the same error in these two series
are different, but you will hardly want to introduce the notion of probability of error in your
exposition. []
I never said that all the numerical results of physics and astronomy are trustworthy, but only
that all the principal results are such. This is indeed very different. []
Letter 3, Markov Koialovich, 1897 (149)
[] Try to eliminate the outlined difficulties not only by unfounded declarations and you
will have to admit that all your calculations were nothing but an obscurity of the problem that
distracted us from the essence of the matter 9.
Letter 4, Koialovich Markov, 5.2.1909 (10, No. 14)
[] That you did not read not only my book, but even its Contents, is obvious since you
discovered an important gap (not a single word is there about the []). This imaginary gap is
filled in in 47, pp. 73 74.
Before advising me to consult generally known textbooks and unnecessarily to reassure
Posse 8, you should have had a look at his course in integral calculus of 1891, pp. 128 189.
[] My definition of the concept of function will hardly mislead those readers who will []
also read note 2 on pp. 39 40. [] 9
1.{See belowNote 5.}
2.{Koialovsky had apparently already read the publications of those two authors.}
3.{The only relevant work is apparently [8]. However, Koialovsky refers to this source
below, and the only possible explanation seems to be that he somehow thought about a few of
Markovs writings on probability in general.}
4.{Cf. Markovs own words which concerned the end of his manual [4, 1908], i.e., the
MLSq [9, Letter 15 of 1910, p. 21]: To my regret, however, I have often heard that my
presentation is not clear enough. On this point, and in connection with Notes 5 and 6 below,
also see [10].}
5.{Koialovich thus distinguishes between ( ) and the density. Anyway, the main point
concerning Markovs manual [4, 1908] is that, when discussing the treatment of observations,
he did not say clearly enough that their errors were random, and that, therefore, they,
practically speaking, possessed some density. At the time, the term random variable (or, in
Russian, random magnitude) was only coming into existence, but at least, beginning with
Gauss onwards, astronomers and geodesists have been discussing random errors.}
6.{In actual fact Markov himself, in the very beginning of his treatise [4, 1908], expressed
his own dissatisfaction with the classical definition of probability.}
7.{Koialovsky did not write out any series (B).}
8.{Konstantin Aleksandrovich Posse (1847 1928), Honorary Member of the Petersburg
Academy of Sciences. His textbook in differential and integral calculus had been widely used
for about 25 years.}
9.{Letters 3 and 4 did not discuss probability.}
(1991) (Scient. Heritage), vol. 17. L.
2. Stokalo, I.Z., Editor (1967),
National Mathematics), vol. 2. Kiev.
(History of
3. 5 . )
.c
(ca. 1915) (New Enc. Dict.), vol. 23, p. 46.
editions, 1908, 1913, 1924; German translation, Leipzig Berlin, 1912.
5. Sleshinsky, I.V. (1893), On the Chebyshev theorem. Zapiski Novorossiysk. Univ., vol. 59,
pp. 503 506 of sect. 1. (R)
6. Yaroshenko, S.P. (1893), On the theory of the method of least squares. Zapiski
Novorossiysk. Univ., vol. 58, pp. 193 208 of sect. 2. (R)
7. --- (1893), Sur la mthode des moindres carrs. Bull. sci. math., sr. 2, t. 17, pp. 113
8. Markov, A.A. (1884, 1885, 1888, 1891),
(Theory of Probability).
Mimeogr. editions. Psb.
10. Sheynin, O.B. (1989), Markovs work on probability. Arch. Hist. Ex. Sci., vol 39, pp.
17. Yu.V. Linnik, N.A. Sapogov, V.N. Tikhomirov
A Sketch of the Works of Markov in the Theory of Probability
In Markov, A.A.
I left aside the first part of this sketch devoted to Markovs work in the theory of numbers.
The authors never mentioned Nekrasov; see P.A. Nekrasov, Theory of Probability. Berlin,
1. Markovs works in this field are mainly devoted to the CLT, as it is now called, for sums
of independent variables; to limit theorems for dependent variables, including variables
connected into a chain pattern introduced by him himself; to urn problems and to issues in
mathematical statistics, and in particular to justifying the MLSq.
2. For ten years, from 1898 to 1908, Markov examined problems connected with the CLT
for independent summands. Overcoming considerable difficulties one after another, and
disregarding the complexity of calculations, he applied Chebyshevs method of moments 1 for
studying the distributions of sums of independent terms and showed that this method was
capable of producing results of an almost exhaustive generality.
At the beginning of the indicated period, A.A. only considered the sums of such independent
terms for which moments of any order existed, but at its conclusion he discovered a way to rid
himself of that burdening assumption and thus to impart the necessary generality to his
findings.
In a letter to Vasiliev 2 of 23.9.1898 Markov [1] examined a sum of such independent
variables X1, X2, , Xn that EXi = 0 and that for any integer k the k-th moment existed and was
restricted in absolute value by a number Ck, generally depending on k, but not on n.
Supposing that Yn = (X1 + X2 + + Xn)/ n, Markov proved that, as n
EYnm Am (EYn2)m/2
t mexp( t2) dt.
Already Chebyshev (1887), studied a similar problem, but his formulations and proofs
contained a gap lacking in the work of Markov.
The proof of (1) was based on the simple properties of the general formula of the Newton
binomial, and of the expectations. It is still important as a component of all his works on the
CLT for independent variables; and he applied its appropriate modifications in his later
contributions devoted to dependent variables connected into a chain or in another way.
3. In his subsequent work Markov achieved success in two directions.
1) He constructed a tool enabling him to determine the limiting distribution of probabilities
for a normed sum by issuing from relations of the type of (1) concerning the limiting behavior
of the expectations of the various integral powers of this sum. This tool was based on the
analytic theory of continued fractions and the related Chebyshev inequalities.
2) He got rid of the restrictions connected with the existence and uniform boundedness of
the moments of high orders by curtailing random variables.
Already in 1883 Markov studied the Chebyshev inequalities in the theory of continued
fractions, and later he explained in detail how to use them in the problem of moments and its
stochastic applications. These studies do not directly belong to probability, but they are
conveniently formulated in stochastic terms.
4. In essence, two different cases were considered, viz., variables having a discrete
distribution or a continuous density. The study of the general case would have led the proof out
of the boundaries of the classical analysis of the 19th century to which he was keeping.
For the discrete case a sequence of random variables X having all the integral
pxk is examined. Then the sum
p/(z x)
for each random variable of the sequence is expanded into a formal continued fraction and the
properties of the (m X) convergents m(z)/ m(z) are established. The denominator has simple
real roots so that
m(z)/ m(z)
[ /(z )]
with an equality valid for any polynomial
( )=
(z) of degree (2m 1):
p (x).
Issuing from this equality and applying very subtle considerations, the following Chebyshev
inequalities are derived. If is situated between two adjacent roots and of the polynomial
m(z), then
If a continuous density f(x) is considered instead of the discrete distribution, then the formally
written integral
f ( x)
should be considered instead of (2) and we derive
In particular, the formal expansion of
[exp ( x2)/(z x)] dx
can be thus examined. We shall have
where and denote the corresponding coefficients and roots. For an unboundedly
increasing m the coefficients and become arbitrarily small.
If now, for each integer k in the sequence Xi of random variables, the relations
x kexp( x2) dx
are obeyed, then, for a fixed m, as it is easily seen,
The difference between the sums in (4) and (5) only consists in the coefficients and for
the roots , or and , and, since and tend to zero as m
, the main relation
exp( t2) dt
for X running through the sequence is thus proved.
5. This provided a full proof of the limit theorem for the case considered by Markov in 1898
when he assumed that all the moments existed and were uniformly bounded and that the ratio
of the variance of the sum to the number of terms was bounded below. Such a theorem was less
general than Liapunovs central proposition, but, in return, the method of proof enabled to
derive relations of the dependence of the approximation of distributions on the approximation
of the moments which are not furnished by the CLT.
Liapunovs fundamental works appeared in 1900 and 1901. There, he applied the method of
characteristic functions rather than the Chebyshev Markov method of moments. For
independent random variables Z1, Z2, , Zn, EZk = ak, var Zk = bk, the existence of bk(2+ ) = E|Zk
bk|2+ was also assumed with > 0 being some constant. For the limit theorem to be valid
only one condition was needed, viz.,
bk(2+ )/(
bk)1+
Subsequent investigations showed that Liapunov had attained almost exhaustive generality.
As Markov [9] indicated, Liapunov had achieved a
generality of conclusions [] far exceeding that secured by the method of expectations. It
seemed even impossible to attain such generality by that method since it was based on
considering such expectations, limitless in number, whose existence in the Liapunov case was
not assumed.
For about eight years, from 1900 to 1908, Markov studied such an application of the method
of moments that would enable him to prove the limit theorem under the Liapunov conditions.
In 1908 he succeeded. The main idea of the new application of the method of moments was the
introduction, along with the main magnitudes Zk ak, of supplementary curtailed random
Xk = Zk ak if |Zk ak| < N, Xk = 0 otherwise.
Such variables already have any moments, and the distribution of their sum when the
number N is chosen appropriately differs but little from the distribution of the initial sum. This
Markovs idea is being successfully applied in stochastic studies until this very day.
6. Markov attached great importance to ascertaining the conditions for the applicability of
the limit theorem for sums of random variables. Even in 1898, in his letters to Vasiliev [1], he
constructed examples of sums of independent random variables which did not tend to the
normal law owing to the comparatively slow increase in the variance of the sum. There also he
criticized Poincars viewpoint on the applicability of the normal law in the theory of
observational errors 3.
Then, Markov [9] offered examples of sums not approaching the normal distribution because
of the violation of the conditions which in present terminology are called after Lindeberg and
Feller.
We also note that Markov studied, although had not solved definitively, the extension of the
limit theorem onto sums of independent two-dimensional vectors. In 1915 he [10] examined
the sums
Sn = (sn1; sn2) =
(xi; yi)
of identically distributed and independent random vectors (xi; yi), i = 1, 2, and proved that
after an appropriate norming the mixed moments of all the orders of sn1 and sn2 tend to the
corresponding mixed moments of the bivariate normal law. The first rigorous proof of the twodimensional limit theorem is due to Bernstein (1944).
7. Markov should be considered the founder of a very important and large section of the
modern theory of probability devoted to studying dependent random variables. The main issues
in which he was interested here related to the LLN and the CLT for sums of random variables.
8. In 1907 4 Markov [2] indicated several sufficient conditions for the LLN to be applicable
to the sums Sn of dependent random variables
Sn =
They were obtained as a corollary of the applicability of the law to any Xs possessing second
moments if only [var (Sn)/n2] 0 as n .
It seems that before the appearance of this work the very possibility of the existence of
sufficiently general conditions for the applicability of the LLN to such sums was not obvious
and Markov [2, p. 361] felt himself obliged to stress that the independence of the variables is
not a necessary condition for the existence of the law of large numbers.
In 1 of his contribution, Markov proved that the law was applicable to Sn if the variances
var (Xi), i = 1, 2, , were bounded and the connection between the Xis was such that the
increase in a particular value of one of them led to the decrease of the expectations of any other
one. At the end of that section he proved that the LLN persisted even when the expectation
E(Xi) of each Xi decreased with the increase in (X1 + X2 + + Xi1).
As a next example of dependent variables obeying the law Markov considered the sequence
X1, X2, ..., Xn, connected, according to modern terminology, into a simple homogeneous
Markov chain with each of the Xis only taking two values, 0 and 1. It was here that A.A. first
introduced such sequences of dependent variables (or dependent trials) which he himself
(although only later) named chains. In essence, the conclusion about the applicability of the
LLN to Sn was obtained here as a corollary of the ergodic property of the simple homogeneous
chain with two possible states that invariably takes place if the transition probabilities differ
from 0 and 1.
In 4 Markov offered an example in which the law was not applicable to dependent and
bounded variables. Its essence consisted in that the law did not apply if |Xi| < L for all values of
i and var Sn > cn2 where the constant c was positive. Finally, at the end of his memoir, Markov
considered variables connected into a simple homogeneous chain with the number of the
possible values of the Xis not being anymore bounded by 2. Once more he obtained the LLN
for Sn as a corollary of the ergodic property of the chain.
9. Again in 1907 Markov [3] proved the applicability of the CLT to the sums Sn of random
variables Xi connected into a simple homogeneous chain and only taking two possible values, 0
and 1, with the transition probabilities differing from 0 and 1.
A.A. indicated that his case offered an example (and, to our knowledge, the first example) of
the applicability of the limit theorem to sums of dependent variables. He obtained the proof of
his proposition by the method of moments basing his reasoning on the symmetry of some
expressions with respect to p and q, the probabilities of the equalities Xi = 1 and 0 respectively.
This symmetry followed from the special conditions of the case under consideration and cannot
be made use of when the variables take more than two values.
10. The work of 1908 [4] is devoted to the study of these more general cases. Here, Markov
began by examining with special thoroughness the assumption that the variables Xi,
constituting the simple homogeneous chain, only take three values, 1, 0 and 1, and modified
the proof applied in his previous work in such a way that it became applicable to a large
number of new problems concerning dependent variables Xi which he examined, one after
another, in later contributions.
It is noteworthy that here, while calculating the moments of the sum Sn, Markov in essence
introduced characteristic functions n( ) = exp[i (Sn an)] of variables (Sn an) where the
constant a was determined under the condition that the expectation E(Sn an) remained
bounded in absolute value as n . Indeed, he began by constructing the function
f (t; z )
n(t)z
which was the generating function of the functions
P(Sn = m)t m,
which, in turn, were the generating functions of the probabilities P(Sn = m). He then changed
the variables, t = eu, z = z1eau, so that the function (5) became the generating function of the
characteristic functions n( ) for imaginary values of the argument :
f ( eu ; z1e au )
au n
n(e )[z1e ]
n(
iu)z1n
eu(man)P(Sn = m) = eaun
iu) =
n(e ).
Romanovsky (1949) later developed in detail the method of introducing characteristic
functions n( ) and applied it to Markov chains properly speaking and their various
generalizations.
At the end of his work Markov extended his result on the applicability of the limit theorem
to sums onto a more general case when variables connected into a chain had any possible
11. As stated above, A.A. also applied the same method in some of his other works. Thus, he
[7] considered the following stochastic pattern: Let some system S randomly take one of its
possible states 1, 2, , k which change one into another at discrete moments of time t1 < t2<
< tn < Suppose that the probability of S being in state i at moment tn depends not only on
its state at the immediately preceding moment tn1 as it happens for simple chains, but on all m
of its previous states tnm, tnm+1, , tn1, although not on n or on the states of S at moments tl, l
> (n m).
If random variable Xn, taking the value xi when the system is in state i corresponds to each
moment tn, then the sequence
is connected, by definition, into a generakized (homogeneous) chain. The problem that
attracted Markov here consisted in studying the sums Sn of such variables under a particular
assumption that m = 2 and that the Xs take only two possible values, 0 and 1. He proved that
under some restrictions imposed on the transition probabilities the CLT is applicable to the
sums Sn. He also indicated an important fact distinguishing generalized and simple chains: for
the former, the variance of the limiting distribution can happen to be identical with that for a
corresponding series of independent variables.
Incidentally, this shows that, given the limiting normal distribution of Sn, it is impossible to
decide whether the Xis were quite independent or in a definite way connected into a
generalized chain.
12. In 1911 Markov [6] solved by means of his method a problem previously considered by
Bruns (1906) from another standpoint. Suppose that variables
Xn = Wn4Wn+1, n = 1, 2,
are the products of two adjacent terms of a sequence
W1, W2, , Wn,
of independent
random variables Wn only taking values 1 and 0 with probabilities and respectively, + =
1. The sequence of dependent variables (iii) represents the simplest case of a Markov Bruns
chain, as it is now called.
The series (iii) is characterized by the property that only the adjacent terms Xi and Xi+1 are
dependent. For |i j| > 1 the variables are independent. This problem prompted Markov to
consider a more general case of arbitrarily distributed bounded variables Xn, n = 1, 2, each
two of which, Xi and Xj, can only be arbitrarily connected one with another if |i j| c where c
was some constant; otherwise he assumed that Xi and Xj were independent. By means of the
method of moments A.A. proved now that the CLT was applicable to the sum (ii) provided that
Bn = var Sn > an where the constant a was positive.
Bernstein (1944) essentially generalized this problem. Applying the method of characteristic
functions he proved that the CLT was applicable to sums of a wide class of such variables Xi
that any of them could be connected one with another although only if the influence of some Xi
on another Xj was, in a sense, sufficiently weak when the distance |i j| was large.
Returning to Markov [6], we note that, in concluding his memoir, he offered some extension
of the simplest case. More precisely, he admitted that the Wn were not independent, but
constituted a simple homogeneous chain.
13. Markov devoted [8] to a problem belonging, in modern terminology, to vector chains.
Consider such a system S with possible states 1, 2, , k that the change from one of them to
another one occurs at discrete moments of time t1 < t2< < tn < and represents a simple
homogeneous Markov chain with transition probabilities having matrix
(P ), i, j = 1, 2, , k.
i j
Let random vectors from an h-dimensional Euclidean space Eh with common possible values
= ( 1(i); 2(i); ; h(i)), i = 1, 2, , k, correspond to moments tn, n = 1, 2, We assume that
vector Xn has value (i) if the system S is in state i at moment tn. The sequence of thus
determined connected vectors X1, X2, , Xn constitutes a vector chain.
The stochastic pattern studied by Markov might be considered a particular case of a vector
chain in the Euclidean space E2. Events A, B, C combined with events E and F constitute six
possible states, AE, BE, CE, AF, BF, and CF. Let vectors Xn = (xn1; xn2) belonging to E2 have
; 1), (
; 0) and (
; 0) respectively
being arbitrary numbers. Markov examined the limiting distribution of the
number of the occurrences of the event E in n trials as n . This problem thus became a
study of the limiting distribution of a sum of random variables
sn2 = (x12 + x22 + + xn2).
The formulation of Markovs several problems differ in the assumptions regarding the matrix
of transition probabilities (iv) and he solved them by his method explicated in detail elsewhere
14. In all the contributions described above, Markov invariably had to do with homogeneous
chains. In 1910 he [5] studied the limiting distribution of the sum Sn of vector variables Xi
connected into a non-homogeneous chain. He assumed that the variables Xi can only take
values 1 and 0 with transition probabilities
p i = P(Xi = 1 / Xi1 = 1), p i = P(Xi = 1 / Xi1 = 0)
and proved that the CLT was applicable to Sn if
po < p i < 1 po, po < p i < 1 po
where po was some positive number. The proof was once more accomplished by the method of
moments, but the moments themselves were calculated quite differently as compared with the
case of homogeneous chains. The manner of calculation can rather be considered similar to that
which he applied for independent variables. It consisted in expanding the calculated
expectation in accord with the Newton formula for a power of a polynomial into separate sums
and isolating their main terms. In the process, A.A. made use of the inequality
var Sn > cn, c > 0, n = 1, 2,
By means of a very clever consideration provided at the end of the memoir, he ascertained
that (7) was always valid if the number po, introduced above, was strictly positive. In 1936
Bernstein, in order to extend the limit theorem onto more general non-homogeneous chains
with the number of possible values of their terms not restricted by 2, had to generalize, first and
foremost, the inequality (7).
We also note that Bernstein provided yet another method of proving limit theorems for nonhomogeneous chains which enabled to generalize essentially Markovs findings. This method
is also applicable to more general kinds of dependent variables and is explicated in detail in his
abovementioned contribution (1944).
15. As a rule, Markov illustrated his general results by random variables occurring when
considering urn patterns, i.e., stochastic schemes realized by some number of urns containing
balls of different colors. The balls served for making trials, for being extracted one by one or
otherwise in accord with precise rules specified in advance. The problem consisted in studying
the limiting distribution of the number of balls of a certain color extracted during n trials as n
. Markov paid attention to such urn patterns not only for the sake of illustration, he also
treated them as a subject in its own right.
16. Many scientists kept to, and developed Markovs ideas concerning chains. Naturally,
homogeneous chains were studied in special detail. The Moscow stochastic school introduced
new important notions here, and Bernstein with his Leningrad students continued to develop
further the theory of non-homogeneous chains. We note that, apparently, especially important
is that development of Markovs ideas about variables connected into a chain which is
contained in the Kolmogorov theory of stochastic processes of the Markov type and the
Bernstein theory of stochastic differential equations.
17. Ideas concerning mathematical statistics occupied an important place in A.As work. We
bear in mind the issues of justifying the MLSq examined in the abovementioned letters to
Vasiliev in 1898 [1] and his treatise [13] as well as his investigation of the coefficient of
dispersion.
When substantiating least squares, Markov in essence introduced new important notions
identical with the now current concept of unbiased and effective statistics for estimating the
constant parameters of laws of distribution when issuing from samples 5. Markov defined them
in common professional terms of observations or measurements corrupted by error. For
substantiating the MLSq he assumed three propositions [1, p. 163] {The authors quote Markov,
see p. of this book}.
He deduced the rule of the arithmetic mean in the simplest case of measuring one magnitude
a when the results of independent observations of equal precision were X1, X2, , Xn and
without introducing any assumptions about the law of distribution of the errors, = X a,
except for the existence of expectation and variance and absence of a constant error (E = 0).
Indeed, when considering linear functions
(unbiased) so that
E( 1X1 +
nXn)
nXn
without a constant error
= a,
it occurs that among these the arithmetic mean, for which i= 1/n, has the least variance.
Later investigations proved that under normality and in the sense of the Markov principles
the arithmetic mean becomes most advantageous not only among all the linear functions of
observation, but also among the wide class of all regular and unbiased statistics.
18. In a more general case m numbers (having no known connections with each other), a1, a2,
, am, are determined, given the values of
bl = yl +
l = 1, 2, , n, n > m
yl = A1(l)a1 + A2(l)a2 + + Am(l)am, i = 1, 2, , m; l = 1, 2, , n,
Ai(l) are known numbers and l are the random errors of observation possessing E l = 0 and var
l = h/pl and pl are the weights of the observations. Here, the same principles are applied. The
numbers al are estimated by linear statistics under unbiasedness and minimal variance. These
statistics coincide with those determined by least squares, hence the justification of the method
from the viewpoint of the Markov principles.
19. Markov [11] studied the coefficient of dispersion by considering series of Si
independent trials in series i (i = 1, 2, , ) where some event E with constant probability P of
occurrence was observed. If Xi is the number of its occurrences in series i, n = S1 + S2 + + S ,
m = X1 + X2 + + X , then the coefficient of dispersion is
Si [( X i / Si ) ( m / n )]2
Q = n(n 1)
( 1) m ( n m)
Markov calculated the expectation and the variance of Q and thus made it applicable for
testing hypotheses on the constancy of P (the hypothesis of homogeneity of the series of trials)
. In a later work he [12] provided the asymptotic distribution of Q as n thus specifying the
test for large values of n.
In these works Markov also criticized the use of the sample correlation coefficient for
estimating the general coefficient when the type of distribution of the general population was
1. {More precisely, the Bienaym Chebyshev method. The same should be borne in mind
in 5.}
2. {Markovs contribution [1] consisted of extracts from letters to Professor Vasiliev
(Kazan).}
3. {Markov had indeed mentioned for example Poincar in connection with the so-called
hypothesis of elementary errors. In actual fact, it goes back to the first half of the 19th century
(Hagen) and even to Daniel Bernoulli.}
4. {Bibliographies state the date of publication of [2] as 1906, but Sapogov (see his Note 2
to [2] in this book) noted that Markov had dated the last part of [2] as 1907.}
5. {The authors could have just as well attributed these concepts to Gauss. In general, their
discussion of Markovs work on the MLSq is faulty; Markov hardly contributed anything to
what was published by Gauss.}
6. {The authors should have mentioned Bortkiewicz and Chuprov if not Lexis, the originator
of the theory of stability of statistical series.}
Except for Ref. [3], the French titles of Markovs contributions are those that he himself
additionally provided for his Russian works. His memoirs [1; 2; 4; 5; 11] are translated in this
1. Markov, A.A. (1899), The law of large numbers and the method of least squares.
2. --- (1907), Extension de la loi des grands nombres, etc.
4. --- (1908), Extension des thormes limites du calcul des probabilits, etc.
5. --- (1910), An investigation of the general case of trials, etc.
6. --- (1911), Sur les valeurs lies qui ne forment pas une chane vritable [14, pp. 399
7. --- Sur un cas dpreuves lies an chane multiple [14, pp. 417 436].
8. --- (1912), Sur les preuves lies en chane par les vnements laisss sans observation
[14, pp. 437 463].
9. --- (1913), The theorem on the limit of probability for the Liapunov cases [14, pp. 319
338]. Translated in P.A. Nekrasov (2004), Theory of Probability. Berlin.
10. --- (1915), Sur lapplication de la mthode des sperances mathmatiques aux sommes
lies. Izvestia AN Psb, ser. 6, t. 9, No. 14, pp. 1453 1484.
11. --- (1916), On the coefficient of dispersion.
12. --- (1920), Sur le coefficient de la dispersion. Note seconde [14, pp. 537 547].
13. --- (1900),
1908, 1913, 1924; German translation: Leipzig Berlin, 1912.
15. Bernstein, S.N. (1944), Extension of the limit theorem of the theory of probability onto
sums of dependent variables. Uspekhi Matematich. Nauk, vol. 10, pp. 65 114. (R)
16. Bruns, H. (1906), Das Gruppenschema fr zufllige Ereignisse. Abh. Schs. Akad. Wiss.
Leipzig, math. phys. Kl., Bd. 29, No. 8, pp. 577 628.
17. Chebyshev, P.L. (1887, in Russian), Sur deux thormes relatifs aux probabilits. Acta
math., t. 14, pp. 305 315.
18. Romanovsky, V.I. (1949), 6
(Discrete Markov Chains). M.
18. A.A. Markov, Jr. The Biography of A.A. Markov
Markov Jr was an eminent mathematician in his own right. He, in his own words, attempted
to describe his father as a man and a citizen but the reader will perceive some inevitable
political undertones; in one case (Note 5) I allowed myself to disagree with the author.
I list more recent work on the same subject in the References to this Foreword, and I mostly
compiled/specified the references to the essay itself.
[1] My father, Andrei Andreevich Markov, was born on 2 June (old style) 1856 in Ryasan.
He was a son of Andrei Grigorievich Markov, an official working at the Forestry Department
and holding the rank of collegiate councillor, then resigning and becoming an attorney, or
solicitor. Andrei Grigorievichs father, Grigori Markovich Markov, was a deacon in the
countryside somewhere near Ryasan 1.
A.Gs retirement was connected with his revealing the abuses practised by the higher-ranked
officials at the Forestry Department. To cover the tracks of their fraud, the influential swindlers
compelled him to retire. A.G. was married twice. By his first wife, Nadezhda Petrovna, a
daughter of an official, he had six children: Petr, Pavel (who died in childhood), Maria,
Evgenia, Andrei and Mikhail; by his second wife, Anna Iosifovna, he had three: Vladimir,
Lidia and Ekaterina.
From among the fathers brothers, Vladimir Andreevich (1871 1897) became well-known
as a first-class mathematician, but tuberculosis brought him to his grave at the age of 26. A
sister of my father, Evgenia Andreevna, was one of the first Russian women doctors. A
psychiatrist by speciality, she worked for a long time in various hospitals for mental patients.
She died in 1920. Also a doctor is Ekaterina Andreevna (born 1875), another sister of my
father now living in Leningrad. {Yet another} sister, Lidia Andreevna, was a teacher in several
high schools. She died in Leningrad in the beginning of 1942. Mikhail Andreevich, a brother of
my father, who died at a great age shortly before the Great Patriotic War {1941 1945}, was
for many years a forester in the Ukraine.
In the beginning of the 1860s, Andrei Grigorievich with his family moved to Petersburg.
Upon his retirement he became the steward of the estate of Ekaterina Aleksandrovna
Valvatieva who had two daughters, Maria (1860 1942) and Elizaveta.
[2] Andriusha Markov was a sickly child. He suffered from tuberculosis of a knee-joint and
walked on crutches. He was nevertheless able to manage without them, but then he hopped on
one leg with the other one never unbending at the knee. Still, he achieved perfection in both
these modes of movement and was even able to play successfully race-and-catch.
When Andriusha was ten, the famous surgeon Kade {Cade? Cadet?} operated on him and
unbended his leg thus enabling him to walk normally. True, he slightly limped all his life, but
this did not prevent him from becoming a good walker and a lover of wanderings. He liked to
repeat the words of a doctor addressed to a postman: You will remain alive as long as you keep
moving.
In 1866 Andriusha entered the 5th Petersburg gymnasium. This classical educational
institution with its red tape was not to the boys liking. All through his life he retained a
gloomy memory of this school whose teachers had tried not so much to teach, but rather to drill
and to dull him. Especially conducive to this aim was the teaching of the classical languages
(Latin and Greek) founded on cramming countless rules and exceptions of all kinds, {rhymed
grammatical sayings}, etc. into the students heads. In most of the subjects, he got on rather
poorly and often brought home unsatisfactory marks. Mathematics appeared to be the only
exception: in this discipline Andriusha invariably got the highest grade. Andrei Grigorievich
once said:
I am not worried about my daughters Masha and Enia {Maria and Evgenia}, they are learning
successfully. But Andrei is giving me great trouble. The Director had again summoned me.
Andrei does not want to study anything except mathematics.
Even as a school student, Andrei was indeed very fond of mathematics, and he studied it on
his own. For some time he thought that he had invented a new method of integrating ordinary
differential equations with constant coefficients. He reported his finding to the well-known
contemporaneous Russian mathema-ticians V.Ya. Buniakovsky, E.I. Zolotarev and A.N.
Korkin. The first did not answer the letter of a school student whereas the other scientists
thoroughly explained him that his method was not really new. Thus my father struck up an
acquaintance with two professors at Petersburg University.
Andrei Grigorievich erred, however, when he thought that his Andrei was not interested in
anything except mathematics. Actually, he was engrossed in the pamphlets of the great public
figures of the 1860s, Chernyshevsky, Dobroliubov, Pisarev, under whose influence the best
part of the student body had been then 2.
In one of his school compositions about {Pushkins} Evgeni Onegin Markov treated it in
Pisarevs spirit for which his teacher vouchsafed him the following remark: You have read a
lot of hack writers who had denied the sense of the beautiful.
An incident nearly resulting in Andreis expulsion from the gymnasium happened during his
study in the graduation class. Once, during the prayer after classes, he was packing up his
books. As ill luck would have it, the last lesson was conducted by the German who zealously
sought for occasions to nag his students. He not so much prayed but rather watched after the
students behavior at the prayer. You are disturbing the reverential feelings of the class, he
declared after the prayer. I shall inform the Director. When Kapustin, Andreis school fellow
who later became a physicist and Professor at Petersburg University, had attempted to plead for
him and told the teacher that the reverential feelings of the class were not disturbed, the teacher
got into a rage and shouted: You are a none advocate 3! Off with you to the punishment cell!
The teacher complained to the Director who summoned Andrei Grigorievich Markov and
declared to him that he will not tolerate atheists or nihilists in his gymnasium. Nothing is
known about the means to which A.G. had to resort to pacify the Director, a three-fold turncoat who changed his religion a few times and finally decided on Orthodoxy.
[3] In 1874 my father graduated from the gymnasium and entered Petersburg University.
There, he attended the lectures of Chebyshev, who influenced his entire scientific work, as well
as those of Professors Korkin and Zolotarev. In addition to reading lectures, the two lastmentioned mathematicians conducted study groups consisting of the best students. My father
participated in them and was quick to solve the difficult problems formulated there. As he
himself owned, his conversations with Korkin were the starting-point of many of his
independent works.
On 31 May 1878 he graduated from the Mathematical Department of the Physical and
Mathematical Faculty of Petersburg University as a candidate {of Sciences}. The same year he
was awarded a gold medal for a composition on a topic proposed by the faculty, On the
integration of differential equations by means of continued fractions, and was left at the
University to prepare himself for professorship. In 1880 he defended his celebrated Master
Dissertation, On the binary quadratic forms of a positive determinant which at once advanced
him to the first ranks of Russian mathematicians. In 1881 he defended his Doctor Dissertation,
On some applications of algebraic continued fractions.
[4] In 1880 he began teaching at Petersburg University as a Privat-Docent. In 1880/1881 and
1881/1882 he read a refresher course in differential and integral calculus, and, in 1883, he took
over the Introduction to Analysis previously read by Yu.V. Sukhotsky and K.A. Posse. Also in
1883, Chebyshev had left the University and my father read, for the first time, a course in the
theory of probability. He invariably read it from 1885/1886.
In 1883 he married Maria Ivanovna Valvatieva. My parents became acquainted already
during their childhood, in the country-house of Ekaterina Aleksandrovna Valvatieva. Later,
when my father became a student, and my mother, a school student with mathematics not
coming easily to her, E.A. invited the stewards son as a mathematics tutor for her daughter.
He soon sought Marias hand in marriage but was not given consent at once. For a long time
my grandmother hesitated being careful not to give away her daughter to a man insufficiently,
as she thought, prosperous and having no definite position. She only assented in 1883, when
my father already was Privat-Docent, about to defend his Doctor Dissertation with the prospect
of a professorship taking shape, and the wedding was celebrated.
On 13 December 1886, in accord with Chebyshevs proposal, my father was elected adjunct
of the {Imperial Petersburg} Academy of Sciences becoming Extraordinary, and then Ordinary
Academician on 3 March 1890 and 2 March 1896 respectively. In 1886 he was appointed
Extraordinary Professor at Petersburg University, and Ordinary Professor in 1893.
He resigned in 1905 remaining Meritorious Professor but continued to read his course in
probability theory. Attempting to find a useful practical application for his main scientific
speciality, he actively participated in the calculations performed at the Pension fund at the
Ministry of Justice both during its establishment and when reviewing its activities.
[5] The scientific significance of my fathers work is discussed in detail in another article of
this collection [5] and I shall not dwell on this subject. I would {rather} like to throw light on
the other aspects of his life and to say what kind of a man and a citizen he was. He was an
open-hearted, frank and bold man. He never betrayed his convictions; all his life he furiously
battled with everything considered by him as stupid or harmful. His civil courage was staunch;
he did not reckon either with the figures against whom he came out or with the possible
consequences for himself. When objections were once raised to one of his suggestions because
of its being in conflict with an Imperial instruction, he answered publicly: I am talking
business but I hear from you nothing except Imperial instruction.
Here are a few facts and documents characterizing my father. It is known that on 25
February 1902 the united sitting of the Branch of Russian Language and Literature and the
Section of Belles-Lettres of the Imperial Academy of Sciences elected A.M. Gorky 4 Honorary
Academician and an official report about this fact appeared in the Pravitelstvenny Vestnik. It
infuriated the Czar Nikolai II who decided to annul immediately the election. To this aim, he,
through the Minister of Peoples Education P.S. Vannovsky, demanded from the Most August
President of the Academy, the Grand Duke Konstantin Konstantinovich Romanov, that the
same Vestnik publish, on behalf of the Academy, an announcement that the election was
annulled. Such an announcement had indeed appeared, although at first (on 10 March) not
from the Academy, but then (on 12 March), at a new request from Vannovsky, on its behalf.
As stated in the announcement, the reason for the annulment was that Gorky was allegedly
under open police surveillance. This was an example of rude arbitrary rule by the autocrat and
by the President of the Academy of Sciences who trembled before him 5 and acted on the
behalf of the Academy without the knowledge or consent of its General Assembly. The
following statement was my fathers response:
To the General Assembly of the Academy of Sciences
I have the honor of proposing to the Assembly that it should insist that the announcement
{itself} about the annulment of the election of Mr. Peshkov to honorary membership be
declared null and void or corrected because, first, it was made on behalf of the Academy
which actually did not annul his election; and, second, the stated reason for the annulment is
insignificant.
6 April 1902. A. Markov
He wanted to read out his statement at the sitting of the General Assembly, but the President
did not permit it and the then Perpetual Secretary, N. Dubrovin, appended an instruction on the
statement: To be left outside the official records. This meant that the statement was not to
appear in the minutes of the Academy of Sciences. Thus the Czarist myrmidons attempted to
deaden any public actions of the scientists directed against the Imperial arbitrary rule.
My father then made the next step: on 8 April 1902 he submitted a request for resignation to
the President. I did not see its text and its very existence is only known from other documents.
The President naturally attempted to conceal it from the public. My father also stated that,
accordingly, he will not further participate in the publication of Chebyshevs {collected}
contributions by the Academy. His resignation was not, however, accepted, and he resumed his
work of publishing Chebyshevs writings being apparently afraid of handing over this
responsible duty to someone else.
In the beginning of 1905 he again protested against the annulment of Gorkys election. He
submitted the following statement to the Academy of Sciences:
Wishing to believe, and to hope that the Imperial Decree of 12 December 1904, that
acknowledged as urgent the taking of real measures for protecting the full force of the law
so that its inviolable and equal for everyone execution would be considered the very first duty
of all the authorities and regions subordinated to us, opens for Russia a new era of legality. I
consider it my duty to recall to the General Assembly the unprecedented case of violating the
law which concerns the Honorary Academician Peshkov who is still not entered in the
academic list and is deprived of the possibility of enjoying the rights of honorary membership.
The annulment of Mr. Peshkovs election to honorary academician was naturally announced in
the newspapers as having been done as though by the Academy of Sciences; but we know that
this announcement was false. Suchlike announcements can be valid only when limitless
arbitrary rule is reigning and fall all by themselves, once the latter is removed.
I am therefore proposing to enter the name of Mr. Peshkov in the list of honorary
academicians and to invite him, in accord with the law {with the by-laws}, to participate in the
life of the Academy.
8 January 1905. Academician A. Markov 6
In accordance with 97 and 98 of the Academys charter this statement was not permitted
to be read out either. Gorky was only invited to a sitting of the Branch of Russian language and
Literature and the Section of Belles-Lettres after the February revolution of 1917 7.
In March 1903 my father submitted the following application to the Board of the Academy
of Sciences:
Since, in accord with the Boards decision, a notification was sent to me about the procedures
for deducting the payment for orders {from holders of decorations}, I have the honor to ask
most humbly to consider that I do not request any orders, nor do I wish to be decorated.
23 March 1903. A. Markov
The essence of this application was naturally not that my father did not want to pay the
deductions for the decorations which were then levied on their holders (the deductions were
insignificant). The essence consisted in that he refused to be further decorated by the Czarist
government and that he disdainfully regarded those orders that he already had. And he took the
first occasion to declare this.
As is well known, on 3 June 1907 the Czarist government dissolved the disagreeable 2nd
State Duma {the Parliament} and issued a new law on electing the 3rd Duma. It thus violated
its own Manifesto of 17 October 1905 according to which it could only have promulgated new
laws with the Dumas consent. And so, my father applied to the Board of the Academy of
Sciences with the following request:
Since the convocation of the 3rd State Duma is connected with a violation of the law, so that it
will not be an assembly of the peoples representatives but some illegal medley, I have the
honor of humbly begging the Board not to enter me in the list of voters. []
S. Oldenburg, the then perpetual Secretary of the Academy of Sciences, appended the
instruction: No action to be taken. In 1908, as a result of the students unrest, the Ministry of
Peoples Education issued a circular letter
where it attempted to put {some} police duties in the hands of university professors. My father
submitted the following appropriate request to the Ministry:
To His Excellency, the Minister of Peoples Education
Concerning the well-known circular letter that was based on an interpretation formulated by
the Senate and shown to me on 25 September in the office of Petersburg University, I consider
it my duty to inform Your Excellency that I decidedly refuse to be a government agent at the
University, although, in accord with the desire of the Physical and Mathematical Faculty, I
reserve for myself the reading of lectures on the theory of probability.
2 October 1908. Academician A. Markov
An appended instruction of the Ministry reads: To be returned to Academician Markov as
improperly submitted.
The Most Holy Synod is known to have excommunicated the great Russian writer, Lev
Tolstoy, from the Russian Orthodox Church. To reveal all the absurdity of this act smacking of
the Middle Ages, my father submitted a request to the Synod for being excommunicated from
the Church as well. Here is its complete text 8.
To the Most Holy Governing Synod
I have the honor of humbly asking The Most Holy Synod to excommunicate me from the
Church. I hope that a reference to my book [6] can serve as a sufficient cause for my
excommunication. There, my negative attitude to the legends underlying the Judaic and the
Christian religions is clearly expressed. Here is a passage from this book (pp. 213 214) {p.
320 in the edition of 1924}: Independently of the mathematical formulas on which we shall not
dwell because we do not attach to them any great importance 9, it is clear that we should
regard stories about incredible events allegedly having occurred in the bygone times with
extreme doubt. And we cannot at all agree with Acad. Buniakovsky [1, p. 326] that it is
necessary to isolate a certain class of stories which he considers it reprehensible to doubt.
In order not to deal with still stricter judges and to avoid accusations of shaking the
foundations, we shall not dwell on this subject that does not directly bear on mathematics.
So that there will remain no doubt about what I am speaking, I adduce the appropriate
passage from Buniakovskys book: Some philosophers, pursuing reprehensible aims, attempted
to apply the formulas concerning the lowering of the probability of testimonies and legends to
religious beliefs and thus to shake them.
If the provided passage is not sufficient, I humbly beg to take into consideration that I do not
see any essential difference between ikons and idols which are of course not gods but their
pictures, and that I do not sympathize with religions which, like Orthodoxy, are supported by,
and in turn lend their support to fire and sword.
12 February 1912. Academician A. Markov
It was not possible anymore to return this request as submitted improperly. The request
caused a terrible commotion in the government camp. The newspapers of the Black-Hundred
orientation raised a wild howl. The Petersburg Metropolitan sent a spiritual pastor, the
archpriest Ornatsky, to admonish and exhort my father who declared, however, to be prepared
only to speak with Ornatsky about mathematical issues, a decision that did not at all suit the
clergyman. The Synod had to comply with my fathers request; in this connection it was
amusingly investigated whether or not he was a sectarian; whether he was baptised; who were
his parents; etc. 11
In 1913, at the governments bidding, the tercentennial of the House of Romanovs was
celebrated. To counterbalance this false Black-Hundred {?} jubilee, my father organized a
scientific anniversary, the bicentennial of the law of large numbers. In one of his letters to
Oldenburg, Aleksei Nikolaevich Krylov 12 wrote about this struggle of my father for justice
with the previous government in the following way [3, p. 320]:
You certainly remember his sharp protest addressed to the Academy about Gorkys exclusion
from the Academy on the instruction of the Minister Sipiagin 13. You apparently also remember
that at the very first sitting of the Academy of Sciences in March 1917 14 Andrei Andreevich
{once more}put forward an unanimously carried proposal to reenter Gorky in the list of
honorary academicians. And how many times did Markov come out at the University with
protests against the measures taken by the previous government and the police with respect to
the University and the student body. For some time he was even discharged from professorship
for these activities. Recall his protest against the Synod concerning Tolstoys excommunication
from the Church; as you know, such protests, always openly and explicitly stated, were
countless, and Markovs very name and the scientific glory certainly gave them strength and
dissemination that were not conducive to the consolidation of the previous government.
[6] My father paid great attention to the teaching of mathematics in the high school. He
vigorously protested against various harmful experiments in this field. In particular, P.A.
Nekrasov, a {former} Professor at Moscow University, attempted to carry out such
experiments. A member of the Black Hundred and a mystic, he aimed at transforming
mathematics into a support for Orthodoxy and autocracy. In 1915, this former warden of an
educational {of the Moscow educational} region, connected with the governing body of the
Ministry of Peoples Education, came out together with P.S. Florov with a plan for introducing
the theory of probability into the school curriculum. Actually, their plan came to the
inculcation of their own confused and pseudo-scientific views on the theory of probability,
mathematical statistics and mathematics in general in the students minds. On my fathers
initiative, the Academy of Sciences set up an ad hoc commission which destructively criticized
that plan [8]. {Accordingly,} it was not implemented, although Florov, who headed a nonclassical high school {Realschule} in Uriupinsk, performed some {appropriate} experiments.
In the 1917/1918 academic year my father himself taught in a high school. [] He solicited
the Academy of Sciences for a yearly trip for continuing scientific work within Russia and we
spent the winter in Zaraisk. In this case, we were my parents, my mothers aunt Serafima
Aleksandrovna Moskvina, and I. I had to transfer from the 5th form of the 8th Petrograd 15
gymnasium to the 6th form of the Zaraisk non-classical school. Mathematics in our class was
taught by the Director, Gilweg, a diligent and pedantic person. He invariably demanded that all
the figures during the lessons in geometry be drawn by ruler and compasses. He suddenly
resigned and the higher forms were left without a mathematics teacher.
To help the school out of trouble and also perhaps satisfying his permanent need to teach
(until then, he had indeed taught for 37 years in succession at the University), my father
offered his free services as a mathematics teacher. His proposal was gratefully accepted and I
thus became his official pupil. His first lesson somewhat frightened and puzzled us. Nothing
resembled the outward carefulness of his predecessor. My father did not even write out the
formulas exactly one under another one. Then {however} his young pupils began to get used to
the Professor (as they called him) but for some time the habits acquired from Gilweg kept
preventing a mutual understanding. Thus, my father once called one of the best school students
to the blackboard and became furious when the pupil, instead of explaining the solution of the
problem, began thoroughly drawing the appropriate figure with ruler and compasses. My father
cried out angrily: Our lesson is in geometry, not mechanical drawing.
He put emphasize on the solution of problems. And he organized additional classes during
vacations and on Sundays for those desiring to advance themselves in this direction. Many
pupils gladly attended and benefited accordingly. The teachers assembly presented him a letter
of thanks for his short but intensive work, and I am still keeping it.
[7] In the fall of 1918 our family returned to Petrograd. My father, who invariably attempted
to do everything in his power for the welfare of his mother country, overwhelmingly longed for
his pedagogic work at the University which then began to return to normal.
In Petrograd he had to undergo an eye operation. For some years, he was suffering from
glaucoma, a grave eye disease, that manifested itself in many representatives of our family at
their old age. Dr. Vygodsky successfully operated my father, his eyesight improved and he
resumed the reading of lectures at the University.
However, the move from Zaraisk to Petrograd told on the general state of his health. In
1920/1921 I walked him arm-in-arm to the lectures which had never been previously needed.
He lectured on the theory of probability conforming to his well-known book. As a listener, I
may testify that, although barely able to keep himself on his legs, he read them faultlessly. At
the same time my father intensively worked on the fourth edition (published posthumously) of
his book [6] which is known to differ considerably from its previous edition.
In the fall of 1921 my father became bedridden. He suffered from a grave kind of radiculitis
accompanied by racking pain. In the spring of 1922 a new disease, aneurism, that developed in
a leg, joined in. Bleeding began. By that time, however, my father became very tired of lying
in bed. He began craving for open air, for being close to nature. The doctor who treated him
permitted a move to the health center for scientists in Detskoe Selo (present name of the city,
Pushkin). This was probably a mistake: the ride in a car unfavorably affected my father and the
bleedings intensified. The head physician of the center decided that an immediate return to
Petrograd for an operation, for the ablation of the aneurism, was necessary.
After the operation my father began feeling himself better and the temperature lowered, but
in a few days there appeared threatening symptoms and {in particular} the temperature rose
sharply. A conference of specialist doctors ascertained a general blood-poisoning and
pronounced his condition hopeless. On 20 July 1922, at 10 p.m., my father passed away. He
was buried in the Mitrofanovsky cemetery in Leningrad.
In 1923 the Academy of Sciences observed the first anniversary of his death. His friend, the
then vice-president of the Academy, Vladimir Andreevich Steklov, vividly addressed the
meeting describing my father as a scientist, a man and a citizen. Thirty years have now passed
since he had died, but Markov the scientist did not, and will not die. His ideas and his
findings, the celebrated Markov chains, the proof of the {generalized} law of large numbers,
the theorems on the minimal values of quadratic forms and his other splendid achievements
were included in the main stock of science and will remain alive for centuries.
1. My fathers biography{written by A.S. Besikovich} included in the posthumous edition of
his textbook [6] mistakenly states that A.G. himself was a deacon. This mistake is repeated in
some of his other biographies.
2.{Nikolai Gavrilovich Chernyshevsky (1828 1889), Nikolai Aleksandrovich Dobroliubov
(1836 1861) and Dmitry Ivanovich Pisarev (1840 1868) were militant writers opposed to
the Establishment and more or less advocating a peasant revolution. Pisarev (cf. below) was
also a staunch antagonist of Pushkins poetry.}
3.{I render the broken phrase that apparently included an intentionally preserved mistaken
loan-translation made by the teacher of German.}
4.{The pen name of the eminent writer Peshkov, as properly named below.}
5.{The trembling is doubtful. The President (born in 1858 and in office since 1889 [4, p.
474]), was about ten years older than Nikolai II (born 1868) and enjoyed more than enough
influence. Note that he had not complied at once.}
6.{I adduce a related letter written by Markov on 23 February 1905 (Archive, Russian
Academy of Sciences, Fond 173, Inventory 1, 60, No. 27) to the President of the Academy of
Your Imperial Highness
On 18 Febr. I received a note signed by Your Highness on 4 February. I have no desire to
criticize it, but the signature of the Most August President of the Academy of Sciences compels
me to make several comments.
And first of all I consider it necessary to state that I cannot change my convictions by order of
the Authorities. Then, as Professor at a university teaching Differential and Integral Calculus
and the Theory of Probability, I ought to turn Your benevolent attention to the fact that in
general the academic institutions are not within the province of the President of the Academy
of Sciences and that only persons fully mastering a certain subject may properly judge the
methods of its teaching.
Finally, perceiving in the note a suggestion to retire, I have the honor to report to Your
Imperial Highness that I shall immediately leave the Academy as soon as its General Assembly
considers my membership superfluous.}
7.{Which preceded the Great October (old style) Socialist Revolution of 1917, as it was
called by the Soviet authorities.}
8. Archive, Acad. Sci. Soviet Union, Fond 173, Inventory 1, No. 65.
9.{In one instance, Markov [7, p. 32] declared that the study of testimonies was the weakest
section of the theory of probability.}
10.{Bearing in mind this date, several authors mistakenly assumed that Tolstoy (who died in
1910) was excommunicated in 1912. Actually, this happened in 1901, but, during Tolstoys
last days, the Synod discussed whether he should be admitted to the bosom of the church.
Nevertheless, he remained excommunicated [9, p. 62n]. This goes to show that in 1912 the
whole story was likely still well remembered but does not explain why Markov had not
submitted his petition two (if not twelve) years earlier.}
11.{Actually, Markov was not excommunicated; the Synod resolved that he had seceded
from Gods Church and expunged him from the lists of Orthodox believers [2, p. 408].}
12. {A naval architect and an applied mathematician.}
13.{Above, in the same connection, the author named another minister.}
14. {Which means: the very first sitting after the February revolution; cf. Note 7.}
15.{The name of Petersburg from 1914 to 1924.}
1. Buniakovsky, V.Ya. (1846), (
(Principles of the Mathematical Theory of Probability). Psb.
2. Emeliakh, L.I. (1954), The case of Markovs excommunication from the church. Voprosy
Istorii Religii i Ateisma, vol. 2, pp. 397 411. (R)
3. Krylov, A.N. (1936),
(Coll. Works), vol. 1, pt. 2. M. L.
4. Levshin, B.V. et al, Compilers (1974), 7
#,
(Academy of Sciences of the Soviet Union, Personnel), vol. 1. M.
5. Linnik, Yu. V. et al (1951), A sketch of the work of Markov, etc. Translated in this book.
editions: 1908, 1913, 1924; German translation: Leipzig Berlin, 1912.
7. --- (1915), On Florovs and Nekrasovs draft. Zhurnal Ministerstva Narodn.
Prosveshchenia, vol. 57, No. 5, pp. 26 34 of the section on current chronicle. (R)
8. Markov, A.A. et al (1916), Report of the Commission to discuss some issues concerning
the teaching of mathematics, etc. Izvestia AN Psb, ser. 6, vol. 10, No. 2, pp. 66 80. Translated
in P.A. Nekrasov (2004), Theory of Probability. Berlin.
Sheynin, O. (1989), Markovs work on probability. Arch. Hist. Ex. Sci., vol. 39, pp. 337
19. Oscar Sheynin.On the Probability Theoretic Heritage of Cournot
. Istoriko-Matematich. Issledovania, vol. 7 (42), 2002, pp. 301 316.
Antoine Augustin Cournot (1801 1877), whose collected works have been recently
appearing, 1 is known as a mathematician, philosopher, economist, educationist and author of
memoirs. Cournots life and work are described by Feller (1961), Weinstein et al (1970) and in
Cournot (1978). In mathematical analysis, he is meritorious for his treatise (1841) reprinted in
1984 with a preface by Dugac who states that, with regard to rigor of explication, Cournot
followed Lacroix and agrees with Weierstrass in that he contributed to the development of the
notion of function. Dugac also quoted Poisson (without, however, providing an exact
reference) who had called Cournot rather an amateur of the highest rank than a geometer in the
strict sense. Somewhat earlier Dugac devoted to Cournot a short note (1978). I remark that
Dugac did not regrettably appraise the methodological value of Cournots treatise (1841).
I mostly discuss Cournots Exposition (1843) 2 in which he treated the theory of probability
and its applications. After Laplace, in 1812, published his Thorie analytique des probabilits,
the need to popularize that discipline began to be felt. Several authors attempted to accomplish
that aim, Lacroix, Cournot, De Morgan, and, in Russia, Buniakovsky. Following Lacroix,
Cournot, as he stated in his Prface, destined his book for readers qui nont cultiv les hautes
parties des mathmatiques and he promised to correct the mistakes and to dissipate the
obscurities left by des plus habiles gomtres.
Cournot had indeed only applied mathematical analysis in some of his footnotes but he thus
did his readers a disservice 3. At the same time, however, in Chapter 12 he applied terms of
spherical astronomy and formulas of spherical trigonometry. His style was ponderous.
Sentences of seven or eight lines and even longer (e.g., in 19, 162, 201, 207) are often met
with, and 117 begins by a phrase of 12 lines. Many elementary calculations (for example, in
13, 70, 165 167, 170, 182, 203, 204) were wrong which, however, did not corrupt
Cournots final conclusions, and, finally, he certainly referred to his predecessors too seldomly
I especially note that Cournot in essence passed over in silence such scholars as De Moivre
and Gauss. Up to the 20th century, mathematicians effectively forgot the latters definitive
approach to the treatment of observations and recalled De Moivres part in proving the De
Moivre Laplace limit theorem even later. Then, although Cournot, in his Prface, cordially
mentioned his late contemporary, Poisson (who died in 1840), and somewhat earlier devoted
his treatise (1841) to Poissons memory, and, moreover, applied the latters term, law of large
numbers, in a previous memoir (1838), he did not say a word about that law in his book 5.
Commentators are unanimous in that the reason for that attitude was the extremely negative
(and nowadays largely forgotten) criticism of the LLN by his excellent ami (Prface, p. 6),
Bienaym (Heyde et al 1977, 3.3).
2. The Main Concepts of the Theory of Probability
2.1. The Theory of Probability. Its aim, as Cournot (1875, p. 181) formulated it, was to
create methods for assigning numerical values to probabilities. He thus moved away from
Laplaces natural-scientific viewpoint (Sheynin 1976, p. 176) who had thought that the theory
was a means for discovering the laws of nature. Recall that Chebyshev (1845, p. 29)
formulated the goal of probability theory as the calculation of the unknown probabilities [of
events] given the probabilities of other [events] and Boole(1851, p. 251) was of the same
opinion, he only replaced Chebyshevs events by propositions.
Cournot (1875, p. 180) had also stated that the tardy origin of the theory of probability was
caused by pure chance, but many later authors including myself (1977, p. 204) have not agreed
with him at all.
2.2. Probability. Cournot (1843, Prface) published Poissons letter of 1836 where that
scholar had explained that he distinguished between the terms chance and probability in the
same way as Cournot did, and that he beaucoup insist [insisted] sur cette difference. A bit
later Poisson (1837, pp. 30 and 31) specified that probability depended on our knowledge and
was therefore subjective, whereas chance was objective. This understanding is now usually
ignored 6. Cournot explained his point of view in 48 rather than in the opening pages of his
treatise and not clearly enough. He insisted, however (see my 5), that probabilities could be
both subjective and objective as well as theoretical and posterior and provided a simple
example (86) in which the probability of a random event could have been only determined
experimentally 7.Finally, he (113) introduced probabilities unyielding to measurement and
(233 and 240.8) called them philosophical. Such probabilities, as he thought, often served
for making judgements. In Russia, Davidov (Ondar 1971), in the mid-century, recognized them
and nowadays philosophical probabilities can be interpreted as expert estimates from whose
treatment modern statistics cannot isolate itself.
Cournot (18) also heuristically defined probability as the ratio of the extent (tendue) of the
favorable chances to that of all of them; a modern definition replaced extent by measure. I
stress that Cournots definition included geometric probability that until him had been lacking
any formula, and thus combined it with the classical discrete case. Newton considered
geometric probability in a manuscript of 1664 1666, Jakob Bernoulli mentioned it, and
Buffon and Laplace applied it; on its history see Sheynin (2003a). The formal birth of the
earlier concept was, however, hardly noticed.
2.3. Density of Distribution. Mathematicians made use of continuous distributions since
Niklaus Bernoulli 8. In 1757 Simpson applied the continuous uniform and triangular laws to
prove that (for them) the arithmetic mean was (stochastically) preferable to an isolated
observation. Cournot (64 65) formally introduced a curve representing the loi des
probabilits des diverses valeurs de la variable 9. Also see his 124 and 125; in 125 he
maintained that the computation of (empirical) tables of probability was comme le chefdoeuvre de la statistique.
In addition, Cournot (73) described the derivation of the density law of a function of a
random variable . Let, in my notation, = f ( ) and 1(x) be the density of . It is required to
derive 2(x), the density of . Then, in 74, he generalized that problem by considering a
function of two variables and examined two examples but did not provide any general
formulas. For that matter, Cournot was mostly interested in such characteristics of the studied
functions as their means, medians (a notion and a term that he himself introduced in 34), as
modules de convergence (126, a statistic inversely proportional to variance). Already Bessel
(1838, 1 2) derived densities of random functions which possibly only remained known to
astronomers. And of course many authors beginning with Simpson studied the stochastic
behavior of the arithmetic mean in the theory of errors.
Following Poisson (1829, 6), Cournot (81) introduced a mixture of densities. Let n1, n2,
observations have densities f1(x), f2 (x), Then
ni fi(x)/ ni
will be the density of the totality of the observations. Cournot had not specified the type of the
component functions but in 132 he referred to his formula and indicated that the numbers ni
could stand for observations made by means of different instruments or under differing
conditions. It is possible that Cournot thus had not excluded an unification of various types of
Here also he was mostly interested not in the function f (x) in itself, but in its characteristics
(cf. above) and his innovation was in agreement with his attempts to treat heterogeneous
observations (cf. my 6.2.2).
2.4. Expectation. Cournot paid little attention to expectation 10 partly because he did not yet
know the Bienaym Chebyshev inequality. He introduced expectation in 50 explaining it in
commercial terms and only provided a numerical example in 61. Further on, when
considering continuous random variables and not daring to make use of integrals, he only
heuristically introduced them (67).
2.5. The Law of Large Numbers. I noted, in my 1, that in 1843 Cournot had not used that
term anymore. He (86) indicated, however, that the Jakob Bernoulli principle allowed to
consider posterior probability instead of the prior and in 115 he called it the seule base solide
for all the applications of the theory of probability (see my 6.2). Many later statisticians
missed his pronouncements and declared that the lack of equally possible cases in practice
meant that it was impossible to apply the theory (Sheynin 2003b, 3.5.3).
Cournot (38) vaguely stated that the application of the theory of probability to the real
world ought to be justified mostly by critique philosophique rather than by raisonnements
mathmatiques. In this connection I note that Chuprov (1905, p. 60 of Russian translation)
attributed to Cournot (1843, Chapter 4) a real substantiation of the LLN. That chapter (which
indeed begins with 38) only describes in detail the notions of randomness (see my 3) and
physical impossibility. Later on Chuprov (1909, pp. 166 168), without providing an exact
reference, stated that Cournots proof of that law was canonical and that the LLN did not
represent either a mathematical theorem or an independent logical principle of an equal worth
with the law of causality. Chuprov first formulated his thoughts about the logical justification
of that law in 1896 when Bortkiewicz regarded them sceptically (Sheynin 1990, pp. 95 97),
and then in 1914. Suchlike reasoning 11 proved unsuccessful: it was the strong LLN that better
described the connection between theory and reality 12.
3. Randomness
Cournot (40) defined randomness 13 as an intersection of independent chains of events. He
thus repeated the interpretation introduced by ancient scholars (Cournot 1843, commentaries
by B. Bru on p. 306). In 45 Cournot mentioned a thorie mathmatique du hazard, calling it
the application plus vaste de la science des nombres [of the widest application of quantitative
knowledge?] whose aim was to study les actes des tres vivants. Bru (Ibidem) noted that
Cournot had not repeated the Laplacean idea (actually due to ancient scholars) of randomness
being caused by ignorance of pertinent causes and remarked that Cournot had attempted to
construct, in the logical and philosophical sense, the calculus of probability on the basis of the
abovementioned theory of randomness. In any case, Cournot (43) connected randomness with
the impossibilit physique, and one of his examples (the impossibility of standing a right
circular cone on its vertex) actually meant that randomness appeared with unstable equilibrium.
Several pronouncements made by later scholars including Maxwell obliquely contained the
same connection between the two notions and Poincar directly defined randomness as a
phenomenon occasioned by small causes leading to considerable consequences.
Cournots further deliberations on physical impossibility had not essentially added anything
to the ideas put forward by Descartes, Leibniz, Jakob Bernoulli and Huygens on moral
certainty (Sheynin 1977, pp. 204 205 and 250 252) whereas Dalembert (Todhunter 1865,
473) directly introduced physical impossibility. I resolutely deny the opinion of some
commentators (Martin 1994) who attribute something new here to Cournot.
Cournot returned to the interpretation of randomness in three later contributions and
considered there mathematical objects. He (1851, 33, Note 38) recalled Lamberts forgotten
attempt to formalize the randomness of the decimal expansions of irrationalities which was
based on his intuitive notion of normal numbers, as they are now called. In accord with his
previous ideas about intersection of chains of events (see above), Cournot also concluded that
there existed no essential difference between randomness and independence of causes.
He (1861, 61, pp. 65 66) then repeated his and Lamberts considerations regarding
randomness (but not his own conclusion), and, finally, he (1879, pp. 177 179) applied
Bienayms sign test of 1874 1875 (Heyde et al 1977, pp. 124 125; Sheynin 1995, pp. 84
85) to checking the randomness of the initial 36 digits of the decimal expansion of (without
pronouncing any definite opinion about that number). In his time, it was impossible to achieve
anything else.
4. Addendum to 3: Natural Selection
Weinstein et al (1970) attributed to Cournot (1851, p. 287/169) the principle of natural
selection. The passage they referred to had consisted of four sentences (one of them of 10, and
another one, of 12 lines) in which Cournot had recognized hereditary variations. He (Ibidem, p.
119/74) also assumed that changes in external conditions led to the extinction of less adapted
individuals, but he (1851, pp. 126 127/78) thought that random causes have played a
secondary part in that process with predestination having been the main agent. Later on
Cournot (1861, 232, p. 362/217; 1872, pp. 155 159/385 390) expressed similar ideas, and,
in the second source, he apparently came to recognize Darwinism in some restricted form.
Finally, Cournot (1875, p. 92ff and p. 103) repeated much of his previous considerations as
well as his partial recognition of the Darwin hypothesis.
The application of stochastic ideas and methods in biology had begun in England at the close
of the 19th century by combined efforts of biologists and mathematicians. Nothing of the sort
happened on the Continent until much later; to some extent this may be explained by Cournots
passive viewpoint, but much more by Quetelets religious attitude. The Belgian scholar, who
died in 1874, had not mentioned Darwin even once; moreover, he (1846, p. 259) declared that
the plants and the animals had remained as created.
5. The Bayes Approach
Cournot (86 95) paid much attention to the rule attribue Bayes (88) and to related
material. This rule, or theorem, as Cournot called it in the sequel, that was nevertheless lacking
in the Bayes memoir (1764 1765), stated that (in modern notation)
P(A/B ) = P(B/Ai)
P(B/AJ)P(Aj).
j =1
The reprint of the first part of the Bayes memoir lacks any indication about its second part.
Nevertheless, the latter indirectly contained a limit theorem. Bayes, as it follows from his note
published in the same source and at the same time as the first part of his memoir, had not
trusted the contemporaneous practice of making use of the sums of the first terms of divergent
series, and apparently for this reason he did not complete his investigation. It was Timerding,
the Editor of the German translation of his memoir, who proved that (Sheynin 1971)
P{ z
( p a)/[ pq /n/n]
z} ~
exp ( x2/2)dx.
Here p and q are the numbers of the occurrence and non-occurrence of the studied event in n
Bernoulli trials (p + q = n ), p is the unknown probability of success in a single trial, a = p/n
= E p and pq/n3 = var p . It is remarkable that Bayes was not satisfied with the finding of De
Moivre
P [ z 8 ( /n p)
z] ~
exp ( x2/2)dx
where was the number of the occurrences of the event in n Bernoulli trials, p and q, the
known probabilities of success and failure in a single trial (p + q = 1) and pqn = var /n. A
modern description of the Bayes memoir is in Hald (1998, Chapter 8).
It was the Bayes approach, first and foremost, that Cournot desired to prove and
correct (cf. my 1). He (89) justly indicated that unsubstantiated (subjective)
assumptions about prior probabilities of the events Ai, i = 1, 2, , n in formula (1) make
objective estimates of the posterior probabilities impossible.
Cournot (95) remarked, however, that, given a large number of observations, probability p
in formula (2) 14 may be assumed equal to the posterior probability p/n. He attempted to
demonstrate this proposition, but hardly successfully. Nevertheless, it is implied by the
Bernoulli LLN and Laplace proved it and indicated a more effective estimate of the rapidity of
the appropriate convergence (Hald 1998, pp. 169 170).
6. Applications of the Theory of Probability
6.1. Astronomy, Mathematical Treatment of Observations, Insurance.
Cournot (145) thought that astronomy was the branch of natural sciences in which the
successful application of probability theory might be expected first of all, and he even
proposed the term statistique des astres. Indeed, the measurement of the proper motions of
stars began in 1837(though at first only perpendicular to the line of vision), and, in the second
half of the century, the regularities inherent in starry clusters came under examination 15.
Cournot himself, however, only studied statistically the parameters of planetary and
cometary orbits. In the former case he (148) assumed a natural supposition about the
probability of a distance between the poles of two randomly situated great circles. Laplace, and
later Newcomb issued, however, from other hypotheses; Bertrand (1888, pp. 6 7) repeated
Laplaces and Cournots reasoning without mentioning anyone and justly noted that the
distinction between the obtained solutions as well as his celebrated paradox concerning the
length of a random chord of a given circle showed that the expression randomly was not
sufficiently definite (Sheynin 2003a).
Somewhat similar happened with the investigation of the cometary orbits. Poisson had stated
that Cournot should have distinguished between two assumptions: either any point of the
celestial sphere was, with one and the same probability, the pole of a cometary orbit; or, all the
inclinations of the orbit were equally probable. Cournot (149) did not agree, also see Brus
commentary on that section.
Cournot described the treatment of observations in more detail than necessary, but not on a
sufficiently professional level, cf. my 1. In particular, he (144) adduced a summary of the
determinations of the length of a seconds pendulum 16, but his inference about the actual
precision of that magnitude, although perhaps correct, was strangely enough based on the
results of its first derivation (by Picard in 1671) 17. It also remains unclear why Cournot had
not discussed meridian arc measurements.
When describing insurance, Cournot hardly applied any new stochastic ideas or methods,
but he had not failed to note that accidents were not independent from each other, cf. my
6.2. Statistics. Applications to statistics are the most interesting in his book. Statistics, he
(103) declared, had blossomed exuberantly and [the society] should be on guard against its
applications prmatures et abusives which can discredit it for some while and delay the time
when it will underpin all the theories concerning the organisation sociale 18. Indeed, already
Lueder (1812, p. 9) complained about the appearance of legions of statistical data.
Statistics, as Cournot (105) continued, ought to have its theory, rules and principles, it
should be applied to physical, natural-scientific, social and political phenomena. He then
formulated a few apparently obvious, but highly questionable at the time statements. The main
aim of statistics was (106) to pntrer autant que possible dans la connaissance de la chose
en soi, and (120) to study the causes of phenomena. The theory of probability was applicable
to statistics although the lment variable escaped our attention [?] and could only be described
by philosophical probabilities (113; see my 2.2). The principe de Bernoulli is the only
foundation for applying probability theory and it satisfies the statisticians (115).
The Staatswissenschaft (Statecraft), a flabby discipline studying political and economic
features of separate states, was not interested in the causes of social phenomena and considered
numerical data insufficient for its purposes. The opposing (sometimes tacit) views of Daniel
Bernoulli, Laplace, Poisson (and Cournot, as seen above), coinciding with the opinion of the
founders of political arithmetic, Graunt and Petty, only became prevailing by the end of the
19th century. Among the statisticians, the originator of the stochastic direction was Quetelet,
who introduced mean (for a given society) inclinations to crime and to marriage, i.e., the
pertinent probabilities. He had not referred to the Poisson LLN (which would have made his
conclusions more general) and did not stress that the calculated inclinations must not be
applied to individuals. No wonder that after his death many statisticians abandoned
Queteletism, and probability theory in general. The LLN did not become an exception, it was
denied on absurd grounds. Knapp (1872, pp. 116 117), for example, even stated that
statisticians always had to do with a single observation, as when counting the population of a
city 19. I dwell now on Cournots concrete ideas and recommendations.
1. The significance of empirical discrepancies. Laplace repeatedly studied the significance of
small discrepancies between observational series (and between theory and observation).
Poisson (Sheynin 1978, 5.2) formally explicated this issue and, like Laplace, restricted his
attention to the case of a large number of observations. Cournot (107 117 and 128 129)
followed Poisson; just as his predecessor (1824, 14; 1829, 1 and 9), he indicated, for
instance in 109, see my 6.2.2, that the probability of an event could be variable. Cournot
(111 and 113) also remarked that apparently considerable discrepancies could contradict
common sense (and be fictitious). However, his refusal to use mathematical formulas (see my
1) made his exposition cumbersome.
2. Series of Bernoulli trials (76 77). Suppose that the series consist of n1, n2, trials
with respective probabilities of success p1, p2, Then, for the totality of the trials, the
probability of success will be
ni pi/ ni.
Denote ni/n = ki, 1 pi= qi, 1 p = q, then
pq = ki pi(1
ki pi).
In quite an elementary way 20 Cournot proved that
pq >
ki pi qi> ( ki pi qi )2.
Commentators (Stigler 1986, p. 197) indirectly indicated that already then formula (3) could
have been applied for justifying stratified sampling.
3. Solidary (systematic) influences and judicial statistics. Cournot (104 and 117) isolated
solidary causes acting in a similar way on a number of trials 21. Random causes, as he
indicated, only corrupted or even determined the results of separate events. This new
characteristic of randomness did not contradict his main definition of that notion (see my 3).
Most interesting was Cournots attempt (Chapter 15), also see the appropriate part of his
paper (1838), to reveal the general causes influencing all the judges (jurors) during a hearing of
a given case 22. Suppose (193) that for two judges the probabilities of a correct judgement are
s1 and s2. Then the probability of the coincidence of their decisions, whether just or otherwise,
will be
p = s1 s2 + (1 s1) (1 s2)
If s1 = s2 = s > 1/2, then (206) s = 1/2 + (1/2) 2 p 1
where p can be determined statistically. If (195), in addition, it were possible to derive, in a
similar way, all the magnitudes included in equations of the type of (4), then a considerable
discrepancy between theory and statistics will testify to an interdependence between the
judges.
Then, again in 206, Cournot recommended to distribute the legal cases among several
heads and form the same equations for each of those. As the number of the categories
increased, then, as he believed, the sources of the miscarriage of justice for each of them will
become independent 23.
I doubt that Cournots recommendations were practicable but in any case he desired to treat
statistical data corrupted by systematic causes, and Bertrands statement (1888, pp. 325 326)
that his predecessor had not noticed the dependence existing between the individual
judgements was absolutely wrong.
Three more points. After Quetelet (and other scholars mentioned in this subsection) it was
Cournot who introduced probability into statistics. Then, he formulated reasonable ideas about
the administration of justice and judicial statistics, and, finally (in 225), he refused to apply
probability for estimating the trustworthiness of a chain of reports. It is not amiss to recall that
Markov (Sheynin 1989, p. 340n) declared that the study of testimonies was the weakest
section of the theory of probability.
4. In my 6.2.1 I noted Cournots qualified remark about possible contradictions between a
formal treatment of statistical data and common sense. Issuing from such reasonable
considerations and in obvious contradiction with what Cournot (113, see my 6.2) had
concluded, Stigler (1986, pp. 197 198) alleged that he had denied the possibility of applying
probability to statistics.
I shall nevertheless adduce two of Cournots critical remarks so as to stress his proper
attitude to statistics. In 102 he indicated that statistical data could be separated in groups in
many ways and that an unfortunate choice of the groups led to mistaken inferences. Then, he
(1851, p. 245) remarked that most often statistical numbers ne mesurer que des effets trs
complexes et trs loigns de ceux quil faudrait saisir pour avoir la thorie rationnelle des
phnomnes but he explained the situation by the dveloppement prodigieux, parfois maladroit
ou prmatur, de ce que lon nomme la statistique, cf. his earlier similar statement in my 6.2.
In his innovations and in the suggested widening of the possibilities of statistics, Cournot
(see my 2.2, 2.3 and 6.2) mostly followed Poisson. This did not depreciate the
methodological importance of his work because the recommendations made by his predecessor
had remained little known. I especially note Cournots efforts to treat dependent statistical
observations and trials with variable probabilities of the studied event; his pertinent
contribution could have considerably fostered the originating theoretical statistics.
Bortkievicz (1894 1896) several times approvingly mentioned Cournot. Still oftener
Chuprov referred to him in his copy of that writing; it is that copy which the translator of
Bortkiewicz had at his disposal. Edgeworth, the predecessor of Karl Pearson, was another
admirer of Cournot. He (1885, p. 571) characterized the latters discussion of posterior
probability as masterly. Elsewhere Edgeworth (1926, p. 446) concluded that Cournot, in his
masterly work on chances, [had] pointed out the bearing of the calculus of probabilities on
statistics.
Nevertheless, here is the opinion of Anderson (1934, p. 539), a student of Chuprov, about
the contemporaneous situation in statistics:
Germanic countries, despite the work of Lexis and his eminent pupil von Bortkiewicz, still
continue under the influence of von Mayrs empirical school. Only in very recent years has
a faint start been made toward the acceptance of the English theories, and, in economic
statistics, of American methods.
Recall also (my 1) that Cournot chose an unfortunate mathematical level for his book and
that his style was flabby. Nevertheless, and the more so since that contribution was translated
into German, it can only be regretted that neither Kries (1886), nor Czuber (1899; 1923), nor
Kaufman (1928) mentioned him at any length. Even Lexis (1879), who studied the stability of
statistical series by issuing from his distribution-free criterion, had not in essence noticed
Cournot. Lexis attempted to determine whether the probability of a studied event varied during
several series of observations, or whether they were dependent one on another. Already
Cournot formulated these aims and at least Lexis could have approached his problems much
earlier than he actually did 24. On the other hand, Chuprov highly (perhaps too highly) praised
Cournot. First, bearing in mind mathematics as well as philosophy and economics, he (1905, p.
60) called Cournot a man of genius. Later on, he (1909, p. 30) stated that Cournot was
One of the most original and profound thinkers of the 19th century, whom his
contemporaries had failed to appreciate and who rates ever higher in the eyes of posterity.
Lastly, Chuprov (1926, p. 77) characterized the French scientist as the real founder of the
modern philosophy of statistics, cf. my 2.5, and this seems to be correct.
1. From 1973 onwards, no less than 13 volumes of his Oeuvres compltes have appeared.
Except for Cournot (1843), which I specify by sections, I provide the page numbers of his
contributions both in accord with their initial editions, and with the OC.
2. Its Russian translation is corrupted by mistakes, misprints and wrong terminology.
Nevertheless, Russian readers might be grateful to the translator, N.S. Chetverikov, a student
of Chuprov, who acquainted himself with Cournots book on his teachers recommendation
(Sheynin 1990, p. 56). Chetverikov was also coauthor of Weinstein et al (1970) where the
French scholar was wrongly credited with the principle of natural selection (see my 4).
3. Cournot repeated Laplaces methodological mistake that his predecessor made in
compiling his Essai philosophique. In 30 he formulated the LLN in words only; in 69, for
instance, instead of providing the appropriate integral, he was compelled to say that a certain
probability was tabulated (at the end of his book).
4. Bru, the Editor of its reprint, meticulously inserted them.
5. Bortkiewicz (Sheynin 1990, p. 42) thought that Poisson was not understood properly; his
statement is not yet examined.
6. Bayes (1764, p. 137), for instance, applied, as he himself stated, both terms, chance and
probability, on a par.
7. Khinchin (1961, No. 1, p. 95) enthusiastically mentioned that example (throws of an
irregular die) but attributed it to Mises. Apparently exaggerating, the latter (1919, p. 57; 1931,
Introduction) called Cournot his predecessor. Actually, however, Cournot (240.5) only hinted
that the limit of the statistical probability might be taken as the theoretical probability, and in a
few instances (see above) stressed, as Jakob Bernoulli did, the importance of posterior
8. He applied the uniform distribution for calculating the mean life of the last survivor from
among a group of men (Todhunter 1865, 340), i.e., the mean value of the appropriate order
statistic. De Moivre derived the normal distribution in 1733, but this fact was forgotten, cf. my
9. Note that already Gauss (1809, 175) introduced the density of distribution in the theory
of errors. Actually following Poisson (1829, 1), modern mathematicians define it as the
derivative of the integral distribution function. The term random variable had not appeared
until the end of the 19th century and possibly even somewhat later (Sheynin 1990, pp. 122
10. He did not at all recognize moral expectation (see his 51 and 61). During its hey-day,
that term became so widespread that Laplace (1812, p. 189) specified the classical term
expectation by calling it mathematical.
11. In attempting to justify the LLN, Ellis (1854, p. 49) remarked that On a long run of
similar trials, every possible event tends ultimately to recur in a definite ratio of frequency.
12. In his Prface Cournot reasoned on the tat pnultime characterizing the behavior of
random phenomena (hasards). Stretching somewhat this idea, it might be interpreted as a
subordination of the variable to the weak (rather than strong) LLN, but it seems better to
disregard that passage altogether.
13. On the history of the concept of randomness see Sheynin (1991).
14. Cournot had not inserted that formula, I am only making use of the appropriate notation.
15. In this connection, as also concerning meteorology, medical statistics and public hygiene
(neither of which Cournot had in essence mentioned), see my papers Sheynin (1982, 1984a;
1984b). Already in 1817, Humboldt had introduced isotherms and thus founded climatology, a
discipline studying the mean values of meteorological elements, after which the study of the
deviations from those means has begun. In medical statistics, a main issue was variolation of
smallpox, a dangerous procedure preceding vaccination.
16. Cournot reduced all these measurements to the latitude of Paris.
17. Cournot (130) reasonably stated that the precision of the final result did not increase
unboundedly with the number of observations, but he (117, 138, 139, 162) only indirectly
and not quite definitely explained the situation. Then, anticipating a resolution of the
International Statistical Congress passed in 1869 (Sheynin 1986, p. 291, Note 18), he (107
and 127) recommended to include in the statistical reports not only the results obtained, but
also their weights.
18. Cf. Cauchy (1845, p. 242): statistics provides a means for judging doctrines and
19. The standard argument was that equally possible cases presumed by the definition of
probability did not exist, and that probabilities were not constant, see my 2.5. The gap
between the theory of probability and the other branches of mathematics, soon beginning to
develop rapidly, was not essential for statistics.
20. In a note attached to 82 Cournot estimated, in a similar manner, the precision of the
function f(x), see my 2.3.
21. Strangely enough, he had not mentioned systematic errors already isolated (although in a
narrow sense) by Daniel Bernoulli in 1780.
22. Keeping to the established tradition (Condorcet, Laplace, Poisson), Cournot both in that
memoir and in the previous chapter of his book first considered the case of independent
judgements. Note that Laplace (1812, p. 523) only once, and in passing, indicated that he
assumed such independence. Other scholars undoubtedly understood that their deliberations
were restrictive and apparently wished to describe the administration of justice at least under
ideal conditions.
23. Bru noted that the last-mentioned statement was arbitrary.
24. Chuprov (1918 1919) proved that the Lexis criterion did not meet the issue;
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20. B.V. Gnedenko. A Review of the Current State
of the Theory of Limiting Laws for Sums of Independent Terms
Uchenye Zapiski Tomsk. Gosudarstven. Pedagogich. Inst.,
No. 1, 1939, pp. 5 28.
Foreword by the Translator
This is an essay on an important issue initially published in a rare source. Regrettably, its
first sections that describe the events in the 19th century are corrupted by mistakes, see my
Note 2, and the type-setter accomplished his work extremely bad. In many cases, I corrected
the formulas but I am not sure that everything is now correct. And, for example, the first term
on the right side of formula (10) is apparently wrong.
1. I review the current state of one of the main chapters of the theory of probability, of the
theory of the limiting laws for sums of independent terms. During the last years a number of
fundamental investigations enriched this chapter. Not only had they given an answer to the
problems raised in the 19th century; they also widely generalized and extended their very
formulation. The research done enables us to say that the main, the principal pertinent issues
are already solved and that a general method, which makes it possible to solve isolated
problems previously treated by most various tricks in one general way and, in addition, to solve
a number of yet unyielding problems, was created in this theory.
Soviet scientists occupy leading positions in developing the theory of limiting laws as also in
furthering other chapters of probability theory. Among those scholars who attained the most
considerable success I ought to name, above all, Kolmogorov, Khinchin and Bernstein. The
students of Khinchin and Kolmogorov, G.M. Bavli, A.A. Bobrov, Gnedenko and D.A. Raikov,
made first-rate discoveries and P. Lvy,and his student Doeblin in France, and Feller in
Sweden, obtained results of no lesser importance, but the basic role belongs to Soviet
mathematicians and this fact will be revealed all by itself in the exposition below.
To make this review intelligible not only for specialists in probability, I provide, in a special
section, brief definitions of such main notions as law of distribution and characteristic function.
And, in pursuing the same goal, I offer a concise description of the process of integration in the
Stieltjes sense. []
4. [] If (x) is a non-decreasing on (a; b) function of bounded variation, then the Stieltjes
integral of a continuous function (x), defined on the same interval, with respect to (x) is the
limit of the sum, as n ,
(xk )[ (xk+1) (xk)]
where xk xk xk+1, x0 = a, xn+1 = b and, as n ,
max (xk+1 xk) 0, 0 k n.
This limit is denoted as
(x) d (x). []
5. Chebyshev [] essentially furthered the study of limiting relations for sums. He had
proved a theorem widely generalizing the LLN [] by a quite elementary but at the same time
very powerful method of moments which later became one of the main methods of the theory
of probability []
A characteristic function of the law F(x) is the function
eixtdF(x) = Eeixt.
f(t) =
Obviously, |f(t)| 1. We will denote such functions by
f(t) = (t) ei
and assume that | (t)| . []
6. At the very end of the 19th century, Liapunov demonstrated a remarkable theorem on the
convergence of the laws of distribution of sums of independent terms to the Gauss law.
The Liapunov Theorem. If independent random terms of the sequence
have finite expectations Exk = ak, variances bk = E(xk Exk)2 and moments dk = E|xk ak|2+
where is a positive constant, and if
(bk ) 2 + ,
then, as n ,
( x k ak )
< x (1/ 2 )
bk ]
( z2)dz.
To prove his theorem, Liapunov had to apply a very involved analytical tool, and, in
essence, to create the method of characteristic functions which became, at the hands of modern
workers, one of the main and most powerful
methods of the theory of probability. Soon, however, in order to rehabilitate the Chebyshev
method of moments, Markov found a new substantiation of the Liapunov theorem which made
use of this method. His approach proved to be quite sufficient for that purpose, but the method
of moments is considerably weaker that the method of characteristic functions which is also
applicable when the random variables do not have finite moments.
Chebyshev, Liapunov and Markov left their contemporaries behind; for more than two
decades their work found no followers either in Russia or abroad. Only in 1922 Lindeberg
provided a new sufficient condition for the sums of independent terms normed by the squared
deviations and centered by the expectations to tend to the Gauss law.
The Lindeberg Theorem. If, for any positive constant $,
(1/Bn2)
(x ak)2 dFk(x) 0 as n
| x a k |>B n
where Fk(x) is the law of distribution of the random variable xk and
Bn2 = (b12 + b22 + + bn2)1/2,
then (2) holds and
(1/Bn2) max bk 0, 1 k n, n .
This theorem was an essential step forward because Lindeberg had not assumed that the
moments higher than the second one existed. In 1935 Feller proved that the Lindeberg
condition was not only sufficient but also necessary for the convergence to the Gauss law. In
1933 Kolmogorov obtained the Lindeberg result anew by the method of differential equations
proposed by I.G. Petrovsky. For a number of problems in the theory of probability this method
proved especially convenient.
7. In 1925, Lvy, in his Calcul des probabilits, further developed the method of
characteristic functions. He expounded it and proved the main theorems concerning the
connection between the behaviour of the characteristic functions and of the corresponding laws
of distribution and applied his theory both for proving the Lindeberg theorem and for
constructing the theory of the so-called stable laws. Lvy formulated and solved the following
problem: Which distributions (x) can serve as limiting laws for the laws of distribution of the
Sn = (x1 + x2 + + xn)/Bn
of independent terms xk, 1 k n, identically distributed in accord with the law F(x) and
normed by appropriately chosen positive constants Bn. It occurred that the class of these
limiting laws coincided with that of stable laws (in Lvys sense). He found a general formula
for them: the logarithm of a characteristic function (t) of a limiting law in the just indicated
sense was expressed as
lg (t) = i"t Co|t| [1 + i t ( )/|t|]
where Co > 0, | | 1, 0 < 2, and " were real constants and
( ) = tg( /2) for ! 1 and = 0 otherwise.
In 1936 Khinchin supplemented Lvys research. He showed that for
law for the laws of distribution of the sums
Sn = [(x1 + x2 + + xn)/Bn] An,
(x) to be a limiting
where An and Bn > 0 were some constants and (1) was a sequence of identically distributed
independent random variables, it was necessary and sufficient that the logarithm of its
characteristics function was the same as (3) but with ( ; t) instead of ( ): ( ; t) = tg( /2)
for ! 1 and = (2/ )lg |t| otherwise.
8. At about the same time (1922 1923), a systematic reconstruction of the theory of
probability on the basis of the theory of functions of a real variable began in Moscow
(Khinchin, E.E. Slutsky). A number of Khinchins contributions devoted to the theory of
limiting theorems for sums have then appeared. In particular, he proved the following
The Khinchin Theorem. Let mutually independent and identically distributed random
variables (1) possess expectations a = Ex1 = Ex2 = = Exn = Then, for any constant > 0,
(xk/n) a| < } 1 as n .
P{|
This theorem completes the investigation of the necessary and sufficient conditions for the
LLN in the Chebyshev form when the terms are identically distributed. Elsewhere Khinchin
proved that in the general case it was impossible to provide necessary and sufficient conditions
only depending on the moments of all integral orders.
Khinchins monograph [4] greatly influenced the taste for investigation in the field of
probability. At the same time, there appeared works that gave an impulse to many later studies
devoted to the so-called strong LLN (Khinchin and Cantelli) 1 and the law of iterated logarithm
(Khinchin). I do not touch these important directions in the development of the classical
theorems of probability theory.
9. In 1927, Kolmogorovs paper devoted to the study of the boundaries of applicability of
the LLN had appeared. There, he offered necessary and sufficient conditions for applying this
law not only in Chebyshevs formulation, but also in a more general setting. Kolmogorov
asked himself, under what conditions was it possible to choose such real constants An, that, for
any constant > 0 and independent random variables (1),
(xk/n) An| < } 1 as n .
In the Chebyshev theorem these constants are equal to the means of the expectations of the
An = (1/n) (Ex1 + Ex2 + + Exn).
Later, in 1937, Feller somewhat generalized this Kolmogorov formulation of the LLN by
studying the conditions to be imposed on the sequence of independent variables (1) so that, for
a given sequence{Bn}of positive constants, it would be possible to choose such real constants
An, that, for any constant > 0,
(xk/Bn) An| < } 1 as n .
In addition to generalizing the formulation of the LLN (above), Kolmogorov, in the same
contribution, essentially changed the very problem for sums of independent terms. Instead of
the classical pattern dealing with a sequence of random variables, he proposed to consider a
sequence of series of random variables independent within each such series, see 10. And his
following theorem that only constituted a particular finding included in his paper, completed
the investigation cycle on the LLN in Chebyshevs formulation.
The Kolmogorov Theorem. For a sequence (1) of independent random variables with
expectations Ex1 = a1, Ex2 = a2, , Exn = an, to obey the relation
(xk ak)|/n] < } 1 as n
P{[|
for any positive it is necessary and sufficient that, as n , the conditions
dFk(x) 0,
(1/n)
| x a k |> n
(x ak) dFk(x) 0,
|x ak | < n
(1/n2) (x ak)2dFk(x) 0
be satisfied.
The following proposition provides necessary and sufficient conditions for a more general
formulation of the LLN.
The Kolmogorov Feller Theorem. For the existence of a sequence of real constants An
such that (5) is valid for any positive where Bn > 0 is a given sequence of constants and (1) is
a sequence of independent variables, it is necessary and sufficient, that, for an arbitrary
constant $ > 0,
dFk(x) 0;
| x |>B n
x2dFk(x) 0.
| x |<B n
It is assumed here that the variables xk are centered by medians:
Fk ( 0) 1/2 < Fk (+ 0).
10. The classical application of the limiting laws in natural sciences consists in the
following. It is supposed that a course of a given phenomenon is influenced by a very large
number of independent random causes each of them little affecting the course as compared
with their combined action. Being added together, the causes provide a resulting (naturally,
random) cause that indeed determines the studied process. Consider now the sequence (1) of
independent variables. Their action is approximately described by the sum
normed and centered by the corresponding constants Bn > 0 and An; that is, by
n=
[(x1 + x2 + + xn)/Bn] An.
As the number of the terms increases, the sum n reproduces the actual phenomenon ever
more precisely. We saw that such a pattern is quite satisfactory for a number of most important
practical issues. []2 However, as Kolmogorov remarked, many natural-scientific problems
did not yield to it, and it was only applied by tradition. In his abovementioned contribution on
the LLN, he indicated a more general scheme that enabled to cover the studied phenomenon in
a much subtler way. He considered a sequence of series
x11, x12, , x1k1 ; x21, x22, , x2k 2 ; ; xn1, xn2, , xnk n
with the random variables being independent within each of these. The phenomenon under
examination is described ever more precisely by the laws of distribution of the sums
Sn = xn1 + xn2 + + xnk n .
Here, we are not anymore interested in the dependence between the random variables of
different series and it is now not necessary to norm and center the sums Sn.
The following general example can illustrate Kolmogorovs pattern. Let us study some
phenomenon depending on the action of random causes. It can be approximately described by
the variables of the first sequence, x11, x12, , x1k1 . We then apply more precise methods of
observation, allow for the previously rejected causes, separate the considered causes into
groups, discard those which do not actually act, and obtain the second series, x21, x22, ,
x2k 2 , for describing the phenomenon. Continuing this process of approxi-mating reality, we
arrive at the next series in (7) and the influence of the separate terms decreases as n increases.
This last-mentioned circumstance, justified by natural-scientific considerations, is thus
formulated in the theory of probability: In the theory of limiting laws for sums, it is thought
that the variables xnk (1 k kn) are asymptotically negligible; that is, uniformly with respect
to k (1 k kn) the relation
P(|xnk| ) 0 as n
holds for any positive constant .
11. For understanding the sequel, we ought to become acquainted with an important class of
the laws of distribution, with the so-called infinitely divisible laws. A law is infinitely divisible
if a random variable subordinated to it can be represented, for an arbitrary natural number n, by
a sum of n independent and identically distributed terms (6).
The development of the theory of time-homogeneous stochastic processes stimulated the
interest to infinitely divisible laws. Bruno de Finetti initiated the study of the theories of both
homogeneous processes and infinitely divisible laws. Only Kolmogorov (1932), however,
attained essential progress here by deriving the formula for the logarithm of a characteristic
function for an infinitely divisible law. He restricted his study to such laws having finite
second moments. Soon Lvy (1934) extended the Kolmogorov formula onto the general case
although without providing a rigorous proof. Khinchin (1937) furnished a full proof of the
Kolmogorov Lvy formula.
As a result of all these investigations, it occurred that for a law (x) to be infinitely divisible
it was necessary and sufficient that the logarithm of its characteristic function be expressed as
iut 1 + u 2
lg (t) = i"t +
[eiut 1
dG(u).
1 + u2 u2
Here, G(u) is a non-decreasing function with bounded variation on ( ; ) and " is a real
It is easy to understand that the Gauss law
(x) = (1/ 2 )
exp (x z2) dz
is infinitely divisible; indeed, in this case we ought to set, in the Kolmogorov Lvy formula,
" = 0 and G(u) = 0 for u < 0 and G(u) = 1 for u > 0.
12. In 1934 1935, Bavli, a student of Kolmogorov, proved the following superb
proposition{s}.
The First Bavli Theorem. If random variables xn1, xn2, , xnk n are independent within each
series,
E xnk n = 0, Ex2nk < C, max Ex2nk 0 for 1 k kn, n
where C is a constant, then the laws of distribution of the sums (6) converge in measure to
some sequence of infinitely divisible laws of distribution. It was Kolmogorov who formulated
the idea about such a convergence of the laws of distribution of sums to a sequence of specially
selected infinitely divisible laws; later he repeatedly expressed the hypothesis that this
convergence takes place always if only the terms xnk are asymptotically negligible.
As a corollary to his general theorem, Bavli obtained a proposition on the convergence of
the laws of distribution of sums to a limiting law:
The Second Bavli Theorem. As n unboundedly increases, the necessary and sufficient
condition for the functions of distribution of the sums (8) to converge to some limiting
distribution function is this: it is possible, under the conditions of the first theorem, to select
such a sequence of constants
n(x)
dFnk( ) +
that, as n , the sequence of functions
tends to some non-decreasing function G(x) at each point of its continuity.
13. In 1936, Khinchin became able to prove a remarkable theorem that determined the class
of limiting laws for sums. It enabled better to establish the part of infinitely divisible laws and
to simplify the proof of some previously established propositions.
The Khinchin Theorem. Each limiting law for a sequence of the laws of distribution of the
sums (8) of terms asymptotically negligible and independent within each series is infinitely
divisible. Inversely, each such law is limiting for some sums of such terms.
Also in 1936 Khinchin formulated the problem about the class of limiting laws for the
classical pattern, i.e., of the laws that can be limiting for the sequence of the laws of
distribution of the sums (4) when the constants An and Bn are appropriately chosen. This class
is naturally included in the class of infinitely divisible laws and Lvy soon exhaustively
described it.
14. We ought to indicate now that essential progress was made in studying the convergence
of the laws of distribution of sums to the Gauss law. In 1935, almost at the same time,
Khinchin, Feller, and Lvy published contributions devoted to this issue. Khinchin provided a
necessary and sufficient condition for the convergence of the laws of distribution of the sums
(4) with appropriately chosen coefficients Bn > 0 and An of independent identically distributed
terms to the Gauss law. Feller discovered necessary and sufficient conditions for the same
convergence when the terms were not identically distributed.
The Feller Theorem. A sequence (1) of independent random variables is given, and it is
required to choose such constants Bn > 0 and An that the laws of distribution of the sums (4)
converge to the Gauss law and that the terms xk/Bn (1 k n) are asymptotically negligible.
For this to be possible, the existence of such a sequence of positive constants Cn that, as n ,
| x |> C n
(1/C2n){
x2 dFk(x) 0,
(1/Cn)
| x |< C n
x2 dFk(x) [
x dFk(x)]2}
is necessary and sufficient.
In the same work Feller proved that the Lindeberg condition was not only sufficient, but also
necessary for the convergence to the Gauss law of the laws of distribution of the sums normed
by the variances
Bn =
E ( xk Exk ) 2
and centered by the expectations An = Ex1 + Ex2 + + Exn.
Lvy also provided a necessary and sufficient condition for the convergence to the Gauss
law; moreover, his formulation was extremely interesting and offered new ideas. His necessary
and sufficient condition for the indicated convergence consisted in that the maximal term
should be asymptotically negligible. At first his formulation was insufficiently clear and the
proof unconvincing, but later he furnished a fundamentally new substantiation with an easily
understandable formulation. Issuing from his result, Khinchin proved the following theorem
ascertaining the basic importance of the Gauss law in the theory of limiting laws.
The Khinchin Theorem (cf. the theorem in 13). If the laws of distribution of sums (8) of
terms independent within each series and asymptotically negligible converge to a limiting law,
then the relation
P(at least one |xnk| , 1 k kn) 0 as n
holding for any > 0 is a necessary and sufficient condition for the limiting law to be
Gaussian.
15. In 1935 Khinchin introduced the notion of relative stability of sums for non-negative
random variables. Suppose that the sequence (1) of independent random variables is given. The
sums (6) are called relatively stable if such a sequence of positive constants Bn can be chosen
that for any positive constant
, as n ,
(xk/Bn) 1| < } 1.
Evidently, relative stability is a natural formulation of the LLN for positive random variables.
At the same time Khinchin discovered a necessary and sufficient condition for relative
stability of sums of identically distributed variables, and indicated that his condition was
analogous to the convergence to the Gauss law in a similar case. Just after Fellers contribution
had appeared, Khinchin formulated the conditions for relative stability in the general case and
his student Bobrov soon proved this hypothetical theorem.
The Bobrov Theorem. A sequence (1) of non-negative random variables is given. For the
sum (4) to be relatively stable, and the terms (xk/Bn), 1 k n, asymptotically negligible, the
existence of such a sequence of positive constants Cn, that, as n ,
xdFk(x)
In a conversation, Khinchin also formulated a proposition similar to the Lindeberg theorem.
I have soon proved it; and, together with Bobrov, we then substantiated it by another method.
In 1938 Khinchins student Raikov formulated this analogy noticed by Khinchin as a first-rate
theorem.
The Raikov Theorem. Let (1) be independent random variables centered by their medians,
Fk ( 0) 1/2 < Fk (+ 0),
and Bn > 0 and An be real constants. For the sequence of the laws of distribution of the sums (4)
to converge to the Gauss law it is necessary and sufficient that the sequence of the sums of the
squares of these variables,
x12 + x22 + + xn2,
is relatively stable.
16. We turn now to considering the sequence (1) of identically distributed random variables
with distribution function F(x). Khinchin proved that for any infinitely divisible law of
distribution (x) it is possible to choose such a sequence (1) and such constants Bn and An that
for some sequence of natural numbers kn the laws of distribution of sums
x1 + x2 + ... + xk n
Sk n =
converge to it. It is said that the law F(x) is attracted to (x). This theorem shows the
importance of a thorough study of the behavior of the laws of distribution for the sums (9).
Lvy, Gnedenko and Doeblin attained a number of pertinent findings.
The totality of the laws of distribution F(x) attracted to a given law (x) is called the domain
of its partial attraction, and if the attraction is restricted by the condition kn = n, we are dealing
with the domain of complete attraction. Understandably, only stable laws possess such a
domain.
It occurred that if F(x) belonged to a domain of partial attraction of a non-stable law, it was
also attracted to a non-countable set of other laws differing one from another. It was also found
that there exist absolutely divergent laws, such that were not attracted to any limiting law
(Lvy, Khinchin, Gnedenko, Doeblin), as well as such (universal) laws which are attracted to
all the infinitely divisible laws (Doeblin). Doeblin and Gnedenko determined the domains of
complete attraction for stable laws.
17. In 1937, issuing from a lemma due to Khinchin, I was able to prove the following
Theorem 1. Given sums (8) of terms asymptotically negligible and independent within each
series. For these sums to converge to a limiting law it is necessary and sufficient that the laws
of distribution n(x) of the sums
S n = x n1 + x n2 + + xnk
of infinitely divisible terms converge to a limiting law. The functions n(x) are determined by
the logarithms of their characteristic functions
lg (t) =
{i nk(1)t + [e 1
dFnk(u) + nk(1)}
where Fnk(x) is the law of distribution of the variable xnk. The limiting laws for both sequences
coincide.
To apply this proposition for deducing limiting laws, we apparently ought to indicate the
conditions for the convergence of infinitely divisible laws.
Theorem 2. For the sequence 1(x), 2(x), , n(x), of infinitely divisible laws to
converge to a limiting law (x) it is necessary and sufficient that the following conditions be
a) Gn(u) G(u) as n at the points of continuity of G(u).
b) The total variations of the functions Gn(u) converge to the total variation of G(u).
c) "n " as n .
The functions Gn(u) and G(u) as well as the constants "n and " are determined by the Lvy
Kolmogorov formula for the laws n(x) and (x).
Since the limiting law for infinitely divisible laws is itself infinitely divisible, Theorem 1
enables to derive as a corollary the Khinchin theorem from 13 on the class of limiting laws.
As a second corollary of Theorems 1 and 2 I indicate the following proposition.
[Theorem 3]. Given, a sequence of sums (6) of independent and asymptotically negligible
terms. For the convergence of their laws of distribution to some limiting law it is necessary and
sufficient that there exist such non-decreasing functions M(u) and N(u) and such constants >
0 and " that
a) M ( ) = 0, N (+ ) = 0,
u dM (u) +
u2dM (u) < + .
b) For x < 0, at the points of continuity of M (x),
Fnk(x) M (x) as n .
c) For x > 0, at the points of continuity of M (x),
[Fnk(x) 1] N (x) as n .
dFnk(x) " as n .
1 + x2
x2dFnk(x) [
| x |<
xdFnk(x)]2}
as n and 0.
The limiting law is determined by the Lvy Kolmogorov formula where " is given by
condition d) and
G(u) =
dM (u), u < 0;
1 + u2
dM (u) + +
dN (u), u > 0.
Theorem 3 enables to obtain, with hardly any work, all the propositions dealt with in this
paper. Theorems 1 3 can thus be considered as the basis of a general method for arriving at
limiting laws for sums of asymptotically negligible terms. The idea of this method is that we
replace the sums of arbitrary independent terms by sums of closely connected with them
infinitely divisible variables. We thus avoid the difficulties encountered when directly dealing
with laws of distribution or when applying characteristic functions, but at the same time make
use of the advantages of both these tools.
The laws of distribution are of a simple functional nature, but for them the addition of
random variables is very complicated. In terms of characteristic functions this operation is
simple, but their analytical nature is very involved. In the method that I propose, the
determination of characteristic functions is reduced, in accord with the Lvy Kolmogorov
formula, to the derivation of a monotonically non-decreasing function with a bounded variation
and the addition of random variables is replaced by the addition of these functions. And, for the
practically most important functions (the Gauss, the Poisson, the stable laws), the functions
G(u) in the abovementioned formula are incomparably simpler than their corresponding laws
of distribution. Owing to these circumstances, my method not only enables to unify the proofs
of isolated (with respect to their derivation) facts, but to decrease essentially the calculations.
Thus, the Feller theorem required lengthy explication (50 printed pages) from its author
whereas its justification by my method is completed literally in a few lines.
18. As an example, I adduce a derivation of the conditions for the convergence of the laws of
distribution of sums to the Gauss law for the Kolmogorov pattern. For the Gauss law the
function G(u) has a single point of growth u = 0 where it experiences a saltus equal to 1; the
functions M (u) and N (u) in Theorem 3 of 17 are therefore identically equal to zero, = 1 and
" = 0. In accord with this proposition, for the convergence of the laws of distribution to the
Gauss law it is necessary and sufficient, as n , that
Fnk(x) 0 for x < 0;
[Fnk(x) 1] 0 for x > 0;
dFnk(x) 0;
xdFnk(x)]2} 1 as 0.
It is easy to understand that these conditions can be formulated otherwise: For the indicated
convergence it is necessary and sufficient that, for any > 0 and n ,
dFnk(x) 0,
xdFnk(x) 0,
xdFnk(x)]2} 1.
We see that the obtained conditions are very close to those put forward by Feller.
19. I complete my paper by a short list of references and indicate some problems which can
be solved by my method (17).
For an initial acquaintance with the theory of probability:
1. Glivenko, V.I.
{New edition: 1939.}
(Theory of Probability), 1938.
For a more detailed study of the modern problems of the theory of limiting laws discussed
above I recommend, in the first place,
2. Khinchin, A.Ya. '
(Limiting Laws for Sums of Independent Random Variables). M. L., 1938.
It is impossible to fail recommending for a most thorough examination two of his other
monographs:
(Asymptotic Laws of the Theory of
Probability). M. L., 1936. {German version, 1933.}
4. (
(Main Laws of the Theory of Probability). M.
L., 1926 and 1932.
Among foreign sources I ought to indicate above all
5. Lvy, P. Calcul des probabilits. Paris, 1925.
6. Lvy, P. Thorie de laddition des variables alatoires. Paris, 1937.
The method expounded in 17 and 18 along with its applications to a number of classical
problems is described in my writing
7. % &'()** +)','-./01 &'()'2 ,-3 4522 /'678*4*201 4-597:/01 8'-*9*/ (On the
Theory of Limiting Theorems for Sums of Independent Random Variables).{Izvestia AN SU,
ser. math., No. 2, 1939, pp. 181 232.}
I do not include{other} papers published in periodicals; a pertinent detailed bibliography is
contained in [6] and [7]; brief bibliographic indications are to be found in each monograph
listed above.
Here, now, are the problems for independent investigation.
1. Theorem 3 of 17 enables to find the necessary and sufficient conditions for the
convergence of the laws of distribution of sums of independent terms to any infinitely divisible
law for the Kolmogorov pattern. For the classical scheme, however, this problem is not yet
solved.
2. The conditions that make it possible to determine whether a given law belongs to the
domain of partial attraction of a given infinitely divisible law are still unknown.
3. Two laws, F1(x) and F2(x), belong to the domain of partial attraction of the law (x); does
the law F(x) = F1(x) * F2(x), that is, the law of distribution of the sum of independent terms
having laws F1(x) and F2(x), belong to the domain of partial attraction of the same law (x)?
4. The properties of universal laws are yet unstudied; neither are the conditions for a given
law to belong to the class of universal laws established.
5. Determine all the laws attracting a given law (Kolmogorov).
1.{The author should have mentioned Borel.}
2.{The author cited the theory of errors which he had indeed mentioned in 4 and 5: I
omitted both these cases; in one of them he (4) wrongly stated that Gauss had justified his
theory of errors in the same way (by the action of a large number of causes, etc). There he also
mistakenly stated that De Moivre had only proved a particular case of the De Moivre Laplace
limit theorem.}
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