text:
\section{ Probabilistic setting and its applications}
 \setcounter{equation}{0}

\noindent Let $\{Z_j,\ j\ge 1\}$ be a sequence of independent integer
valued random variables that induces a sequence of random vectors
$\{{\bf K}^{(n)}=(K_1^{(n)},\ldots,K_n^{(n)}),\ n\ge 1\}$ given by

\begin{equation} {\cal L}({\bf K}^{(n)})= {\cal L}(Z_1,\ldots, Z_n \vert
\sum_{j=1}^njZ_j=n), \quad n=1,2,\ldots. \label{1}
\end{equation}

\noindent It follows from \refm[1] that ${\bf K}^{(n)}\in \Omega_n, \
n\ge 1,$ where \begin{equation} \Omega_n=\{\eta=(k_1,\ldots, k_n):\sum_{j=1}^n
jk_j=n \}
\end{equation}
\noindent is the set of all partitions $\eta$ of  an integer $n.$
 \noindent In probabilistic combinatorics, \refm[1] is called
the conditioning relation (see\cite{ABT}), while the sequence of
vectors $\{{\bf K}^{(n)},\ n\ge 1\},$ is called the counting
process.

\noindent Next, denote by $\mu_n$ the probability measure on $\Omega_n$
induced by the conditioning relation \refm[1]:$$\mu_n(\eta):=\bbb P
({\bf K}^{(n)}=\eta),\quad \eta\in \Omega_n,\quad n\ge 1$$ and let
\begin{equation} a_k^{(j)}=\bbb P (Z_j=k), \quad k\ge 0, \quad j\ge 1. \label{2}
\end{equation}
\noindent We then have \begin{equation} \mu_n(\eta)=c_n^{-1} \prod_{j=1}^n
a_{k_j}^{(j)}, \quad \eta=(k_1,\ldots, k_n)\in \Omega_n, \quad
n\ge 1, \nonumber
\end{equation}

\noindent where \begin{equation} c_n=\bbb P(\sum_{j=1}^n jZ_j=n)=  \sum_{\eta\in
\Omega_n} \prod_{j=1}^n a_{k_j}^{(j)}, \quad \eta=(k_1,\ldots,
k_n)\label{3}
\end{equation}

\noindent is the  partition function for the measure $\mu_n.$ We will
assume throughout this paper that the probabilities $a_{k}^{(j)},\
k\ge 0, \ j\ge 1$ \noindent are such that  $c_n>0,\ n\ge 1$.

 \noindent It is clear that the sequence of measures $\{\mu_n, \ n\ge
1\}$ induced by \refm[1]
 is uniquely defined by
the array of probabilities $\{a_j^{(k)},\ j\ge 1,\ k\ge 0\}.$
However, this correspondence is not a bijection. In fact, the
``tilting" transformation (see \cite{ABT}) of the probabilities:
\begin{equation} a_k^{(j)}(\rho)= \frac{\rho^{jk}a_k^{(j)}}{S^{(j)}(\rho)},\
k\ge 0,\ j\ge 1,\label{ti}\end{equation} where $\rho> 0$ and
$S^{(j)}(\rho)$ is the normalizing constant, does not change the
sequence $\{\mu_n\,\ n\ge 1\}$. But this transformation does
affect the partition function $c_n:$ \begin{equation} c_n(\rho)=\frac{c_n
\rho^n}{ \prod_{j=1}^nS^{(j)}(\rho)}, \quad n\ge
1.\label{zz}\end{equation} Note
 that the tilting  is defined for all finite $\rho> 0,$ such that
\begin{equation} S^{(j)}(\rho)=\sum_{k=0}^\infty\rho^{jk}a_k^{(j)}<\infty, \
j\ge 1.\label{sj}\end{equation}

 \vskip .5cm It is a remarkable fact  that
the representation \refm[1] provides a mathematical formalism for
a variety of models in seemingly unrelated contexts. Let us
briefly describe    four main  fields of application of this
setting.

$\bullet$ {\bf Decomposable combinatorial structures} (for more
details see \cite{ABT},\cite{A},\cite{Kol}). We consider
combinatorial structures. The size of such a structure is defined
to be the number of elements in it. A decomposable structure of
size $n$ is a union of indecomposable
 components, so that the counts $k_1,\ldots ,k_n$ of
components of sizes $1,\ldots,n $ respectively, form an integer
partition of $n.$ It is  assumed that each  component of size $j$
belongs to one of  $m_j$ types. The
 three classes
of decomposable structures: assemblies, multisets and selections,
encompass the whole  universe of classical combinatorial objects.
Assemblies are structures composed of labelled elements. The class
of assemblies  includes permutations  decomposed into cycles
($m_j=(j-1)!$), forests composed of rooted trees with labelled
vertices
  ($m_j=j^{j-1}$), graphs composed of connected subgraphs with
labelled vertices ($m_j\sim 2^{{j\choose 2}}$), etc. Multisets are
formed from unlabelled elements. Examples of this class of
structures are integer partitions ($m_j=1$), planar partitions
($m_j=j$) and mapping patterns ($m_j\sim \frac{\rho^{-j}}{2j},\
 \rho=0.3383$). Finally, selections are defined as multisets with distinct
components, which means that all component counts $k_j, \
j=1,\ldots,n$ are either $0$ or $1.$ A typical example of a
selection is an integer partition into distinct parts ($m_j=1$).

The basic problem in enumerative combinatorics is to find the
asymptotics, as $n$ goes to infinity, of the number of a certain
class of structures  of size $n$, having a given component
spectrum $(k_1,\ldots,k_n)\in \Omega_n.$ As a part of this
question, the asymptotics of the total number of given structures
of size $n$ is of special interest.

The starting point of the probabilistic method considered is the
definition of  a random structure of size $n$ which is a random
element $\Pi_n$ distributed uniformly on the finite set of all
given structures with  size  $n$. Next it is defined the induced
random component spectrum (also called counting process)
$${\bf K}^{(n)}= {\bf
K}^{(n)}(\Pi_n)=(K_1^{(n)},\ldots,K_n^{(n)}),\ n\ge 1,$$ where the
random variable $K_j^{(n)}$ represents the number of components
of size $j$ in  $\Pi_n$.

It turns out that
 the representation \refm[1] of the distribution of ${\bf K}^{(n)}$
is valid for the aforementioned three classes of  combinatorial
structures. Namely, assemblies, multisets and selections are
induced respectively, by the following
 three types of random variables $Z_j, \ j\ge 1:$
Poisson $(Po(a_j), \ a_j={m_j\over j!})$, Negative Binomial
$(NB(p_j,m_j))$ and Binomial $(Bi(p_j,m_j))$.
\\
$\bullet$ {\bf  Models of ideal gas} (for references see
\cite{V1},\cite{Grei}, Ch.12, \cite{Sal}).

An ideal gas, treated as a quantum system, is a collection of
perfectly elastic particles (atoms or molecules) which collide but
otherwise do not interact with each other. It is assumed that the
total internal energy $E$ of a gas is the microscopic energy of
the random motion of individual particles and that $E$ is
partitioned between the particles, so that $k_j,$ called an
occupation number, is the number of particles with energy value
(energy eigenvalue) $\epsilon_j$, i.e. \begin{equation} E=\sum_{j\ge 1}
\epsilon_j k_j. \label{E}\end{equation}
 In the case of a $d$- dimensional ideal gas, a particle is viewed as a lattice point
  ${\bf q}\in{\cal Z}_d$ and the energy values are of the following special
  form:
  \begin{equation}\epsilon_{\Vert \bf{q}\Vert^2}=c\Vert {\bf q}\Vert^2,
  \ {\bf q}=(l_1,\ldots,l_d)\in {\cal Z}^d, \quad \Vert
  {\bf q}\Vert^2=\sum_{s=1}^d l_s^2,
  \label{eps}\end{equation}
where  $c>0$ is a known constant that does not depend on $\bf{q}.$

 Now we see from \refm[E],\refm[eps] that, given a total energy
$E$, a state of an ideal gas is determined by a partition
$\eta=(k_1,\ldots,k_n)$ of an integer $n.$ The probability
distribution of the energy states $\eta$ which varies from model
to model, is defined by a measure on the state space $\Omega_n$.
By laws of statistical mechanics, these measures are forced to be
of the form (\ref{3}), with the numbers $a_k^{(j)}$ defining the
type of a model of ideal gas considered. Vershik (\cite{V1})
suggested to call the class of measures (\ref{3}) multiplicative.
Observe that the multiplicative form \refm[3] of the sequence of
measures $\mu_n$ is implied by the fact that the random variables
$Z_j, \ j\ge 1$ in \refm[1] do not depend on $n.$

 The following three basic models (statistics) of
an ideal gas are distinguished: \\ Maxwell-Boltzmann(MB) (labelled
particles, without restrictions on values of occupation numbers),
Bose-Einstein(BE) (indistinguishable particles, without
restrictions on values of occupation numbers) and Fermi-Dirac(FD)
(indistinguishable particles, such that no more than one particle
may occupy a given energy level). In full similarity to the
setting for combinatorial structures, MB, BE and FD statistics are
induced by the same  three types of random variables $Z_j$ in
\refm[1]: $Po(a_j)$, $NB(p_j,m_j)$ and $ Bi(p_j,m_j)$
correspondingly. In the case of  the classic $2$- dimensional MB,
BE and FD statistics, the parameters $m_j=const, \ j\ge 1$, while
to the parameters $p_j$ and $a_j$ it is attributed the meaning of
the average of the occupation number at energy level $j$ which has
a known in quantum mechanics expression via the corresponding
energy eigenvalue $\epsilon_j$. This says that the classic BE and
FD models conform respectively to geometric and bernoulli
distributions of $Z_j, \ j\ge 1$, which  induce the uniform
probability measures $\mu_n$ on the set of all integer partitions
of $n$ and on the set of integer partitions of $n$ with distinct
summands, respectively.

 In the case of a $2<d$- dimensional ideal gas,
the situation is more complicated. By \refm[E], \refm[eps], to
each energy level $j$ is naturally prescribed a "weight" $r_d(j)$
which is the number of representations  of the natural number $j$
as a sum of $d$ integer squares. In other words, $r_d(j)$ is the
number of distinct lattice positions ${\bf q}=(l_1,\ldots,l_d)\in
{\cal Z}^d$ of a particle on a sphere of radius $\sqrt j,$ i.e.
 with the energy level $j.$ It is known from  number theory (see
e.g. \cite{g}) that for $d\ge 5,$ $$C_1j^{\frac{d-2}{2}}\le
r_d(j)\le C_2j^{\frac{d-2}{2}},\ j\ge 1, $$ where $C_1,C_2$ are
positive constants depending on $d,$ and  that for $d=2,3,4$ the
functions $r_d(j)$ oscillate wildly (in $j$), while, obviously,
\begin{equation} r_1(j)=
  \begin{cases}
    2 , & \text{if  $j$ is a square} \\
    0 , & \text{otherwise}.
  \end{cases}
  \end{equation}

Employing  known  properties of   $r_d(j),$ an important fact was
proven in \cite{V2} that
 for the sake of asymptotic analysis, it is possible to treat
 the $d$- dimensional BE and FD models as the classic ones
 with parameters $m_j=cj^\beta,\ \beta>-1, \ c>0,$
where $\beta=\frac{d-2}{2}, $ if $d\ge 2$ and
$\beta=-\frac{1}{2},$ if $d=1.$

$\bullet$ {\bf  Coagulation--fragmentation processes on the set of
integer patitions} (see \cite{DGG},\cite{ABT}).

 \noindent  We will show  that a multiplicative measure
$\mu_n$ can be viewed as an equilibrium of a classic
coagulation--fragmentation process (CFP) which is a
time-continuous Markov chain on $\Omega_ n$, defined as follows. A
state $\eta=(k_1,\ldots,k_n)\in\Omega_n$ of a  CFP depicts a
partition of a total number $n$ of identical particles (animals,
atoms, stars, human beings, etc) into clusters (groups) of
different sizes, so that $k_j$ is the number of clusters of size
$j.$ The only possible infinitesimal (in time) transitions are
coagulation (merging) of two clusters of sizes $i$ and $j$ into
one cluster of size $i+j$ and fragmentation (splitting) of a
cluster of size $i+j$ into two clusters of sizes $i$ and $j.$
Given a state $\eta\in \Omega_n$ with $k_i,k_j>0$ for some $1\le
i,j\le n,$ denote by $\eta^{(i,j)}\in\Omega_n$ the state that is
obtained from $\eta$ by the  coagulation of any two clusters  of
sizes $i$ and $j$, and denote by $u_c(\eta,\eta^{(i,j)})$ the rate
of the infinitesimal transition $\eta\rightarrow \eta^{(i,j)}.$
Similarly,
 for
a given state  $\eta\in\Omega_n$ with $k_{i+j}>0,$ let
$\eta_{(i,j)}$ be the state that is obtained from $\eta$ by the
fragmentation of any cluster of size $i+j$ into two clusters of
sizes $i$ and $j,$ and let $u_f(\eta,\eta_{(i,j)})$ be the rate of
the infinitesimal transition $\eta\rightarrow \eta_{(i,j)}.$
Denoting by
$$q(\eta;i,j)=\frac{u_c(\eta,\eta^{(i,j)})}{u_f(\eta^{(i,j)},\eta)}$$
the ratio of the above transitions, the  important property of
reversibility of multiplicative measures is derived by verifying
 the detailed balance condition.

\noindent \begin{proposition} A multiplicative measure $\mu_n$ defined
by \refm[3] is reversible with respect to the  transition rates
$u_c,u_f,$ such that their ratio satisfies:
 \begin{equation}
 q(\eta;i,j)=
  \begin{cases}
    \frac{a_{k_i-1}^{(i)}a_{k_j-1}^{(j)}a_{k_{i+j}+1}^{(i+j)}}
{a_{k_i}^{(i)}a_{k_j}^{(j)}a_{k_{i+j}}^{(i+j)}},&\quad
   \text{if}\ \ i\neq j:k_i,k_j>0\\\\
       \frac{a_{k_i-2}^{(i)}a_{k_{2i}+1}^{(2i)}}
{a_{k_i}^{(i)}a_{k_{2i}}^{(2i)}},&\quad \text{if}\ \  i=j: k_i\ge
2.
  \end{cases}
\label{cv}
  \end{equation}

\end{proposition}
\vskip 1cm  An immediate consequence  of  the Proposition 1 is
that a multiplicative measure $\mu_n$ defined by \refm[3] is the
equilibrium distribution of a CFP with transition rates obeying
the condition \refm[cv].

We now distinguish a class of CFP's with transition rates of the
form  $$ u_c(\eta,\eta^{(i,j)})=
  \begin{cases}
    k_ik_j\phi(i,j) , & \text{if}\quad i\neq j, \quad k_ik_j>0 \\
    k_i(k_{i}-1)\phi(i,i) , & \text{if}\quad i=j, \quad k_i\ge 2,
  \end{cases}
$$
\begin{equation} u_f(\eta,\eta_{(i,j)}) =k_{i+j}\varphi(i,j), \quad 2\le i+j\le
n,\quad k_{i+j}\ge 2, \label{qw}
 \end{equation}
where $\phi,\varphi $ are some symmetric nonnegative functions on
the set of pairs of positive integers. Treating   the functions $
\phi,\varphi $ in \refm[qw] as the rates of a single coagulation
and a single
 fragmentation respectively, the induced CFP's can be viewed as  mean-field
 models on the set $\Omega_n$. In fact, \refm[qw] tells us that at any state $\eta\in \Omega_n,$
each cluster can coagulate with each other one or can be
fragmented into  two parts, so that the net rates of the
transitions $\eta\rightarrow \eta_{(i,j)}$  and $\eta\rightarrow
\eta^{(i,j)}$ are  sums of the rates of all possible, at the state
$\eta$, single coagulations and
 fragmentations respectively. It is proven in \cite{DGG} with the
 help of Kolmogorov cycle condition that in the case when all rates of single
  coagulations  and fragmentations
   are positive,
 the  CFP's given by
 \refm[qw] are reversible if and only if the ratios of the single
 transitions are of the form
\begin{equation}\frac{\phi(i,j)}{\varphi(i,j)}= \frac{a_{i+j}}{a_{i}a_j}, \quad
i,j\ge 1,\label{143}\end{equation} with some  $a_i>0, \quad i\ge 1.$
The corresponding CFP's are known as classical reversible models
of clustering and networks studied in 1970-s by Kelly and Whittle
(see \cite{DGG}-\cite{EG} and references therein). The equilibrium
measures of the mean field CFP's with the rates \refm[qw],
\refm[143] are multiplicative measures $\mu_n$ induced by the
conditioning relation \refm[1] with $Z_j$ distributed $Po(a_j), \
j\ge 1$. A clear example of a reversible  CFP which is not a mean
field, is provided by letting  $Z_j=G_j-1$, where $G_j$ is
distributed geometrically ($p^j,\ 0<p<1$). Then
$a_{k}^{(j)}=p^{jk}q_j, \ q_j=1-p^j, \ j\ge 1, \ k\ge 0 ,$ and the
corresponding measure $\mu_n$ is a uniform one on the set
$\Omega_n,$ while reversibility holds iff $q(\eta;i,j)\equiv 1,$
by \refm[cv]. Simple calculations demonstrate that, due to the
last fact, the representation \refm[qw] of net transition rates
fails for the CFP considered.

 \noindent $\bullet$ {\bf CFP's on set partitions}(\cite{BP},\cite{P},
 \cite{Bert}).
We assume here that in the preceding set up for CFP's, particles
are labelled by $1,\ldots ,n,$ so that  the state space of the
system of clusters related to some CFP, becomes the set
$\Omega_{[n]}=\{\pi_{[n]}\}$ of all partitions $\pi_{[n]}$ of the
set $[n]=\{1,\ldots,n\}.$ Recall that a partition of $[n]$ into
$k$ blocks (clusters) $A_1,\ldots, A_k$ is
$\pi_{[n],k}=(A_1,\ldots, A_k),$ where $A_j, \ 1\le j\le k\le n$
are nonempty and disjoint subsets of $[n]$ whose union is $[n]$
and which are numbered, e.g. in the order of their least element.
Denoting $\vert A_j\vert$ the size of a cluster $A_j,$ we further
assign to  each $A_j,$ a weight $m_{\vert A_j\vert} $ which is a
number of  possible states of $A_j$, the states can be e.g.,
shapes (in the plane or in space), colors, energy levels, etc.
This says that to the set partition $\pi_{[n],k}$ correspond
$\prod_{j=1}^k m_{\vert A_j\vert} $ different structures with the
same blocks $A_1,\ldots, A_k$, so that the total number of
structures formed by all partitions of the set $[n]$ into $k$
clusters is equal to \begin{equation} \sum_{\pi_{[n],k}\in
\Omega_{[n],k}}\prod_{j=1}^k m_{\vert
A_j\vert}:=B_{n,k},\label{Bnk}\end{equation} where $B_{n,k}$ is
known as a Bell polynomial in  weights $m_1,\ldots, m_{n-k+1}.$
Similar to the above setting for decomposable combinatorial
structures,  a random structure $\Pi_{[n],k}$ is the one chosen
randomly from the set of $B_{n,k}$ structures. As a result, for a
given $k$ a measure $p_{[n],k}$ on the set
$\Omega_{[n],k}=\{\pi_{[n],k}\}$ is induced: \begin{equation}
p_{[n],k}(\pi_{[n]})=\frac{\prod_{j=1}^k m_{\vert
A_j\vert}}{B_{n,k}},\quad
\pi_{[n]}\in\Omega_{[n],k}.\label{gb}\end{equation} In a more
general setting which encompasses a variety of models (see
\cite{BP},\cite{P}), the weights $m_j$ in \refm[gb] are allowed to
be arbitrary nonnegative numbers. Pitman \cite{P} calls the
$\Pi_{[n],k}$ a Gibbs partition and the measure $p_{[n],k}$
microcanonical Gibbs distribution. Obviously, the vector $(\vert
A_1\vert,\ldots \vert A_k\vert)$ of block size counts defines a
partition of the integer $n$ into $k$ summands induced by the
generic  set partition $\pi_{[n],k},$ and it is known that  to
each $\eta=(k_1,\ldots,k_n)\in \Omega_n,$ such that
$k_1+\ldots+k_n=k,$ correspond
$$\frac{n!}{\prod_{j=1}^n (k_j!)(j!)^{k_j}}$$
different partitions $\pi_{[n],k}\in \Omega_{[n],k},$ each one of
them having   the same probability $p_{[n],k}(\pi_{[n],k}),$ given
by \refm[gb]. Thus, the Gibbs distribution $p_{[n],k}$ on
$\Omega_{[n],k}$ induces the Gibbs distribution $p_{n,k}$ on the
set $\Omega_{n,k}$ of integer partitions of $n$ into  $k$ positive
summands: \begin{equation}
p_{n,k}(\eta)=(B_{n,k})^{-1}\prod_{j=1}^n(\frac{m_j}{j!})^{k_j}\frac{1}{(k_j)!},
\quad \eta=(k_1,\ldots,k_n)\in \Omega_{n,k}, \label{pnk}
\end{equation}
where the partition function $B_{n,k}$ defined as in \refm[Bnk]
can be rewritten in the following form: \begin{equation} B_{n,k}=\sum_{\eta\in
\Omega_{n,k}}\prod_{j=1}^n(\frac{m_j}{j!})^{k_j}\frac{1}{(k_j)!}.
\end{equation}
 From \refm[pnk] it is easy to derive   Kolchin's
representation of Gibbs partitions (see \cite{P}, Theorem 1.2).
From the other hand, the distribution $p_{n,k}$ given by
\refm[pnk] is produced by conditioning the multiplicative measure
measure $\mu_n$ given by \refm[2], on the event $Z_1+\ldots
+Z_n=k,$ with $Z_j\sim Po(a_j), \ a_j=\frac{m_j}{j!},\ j\ge 1.$
However, this embedding of the generic model associated with set
partitions of $[n]$ into the setting for conditioning relation
\refm[1] does not facilitate the study of a wealth of problems
(see \cite{BP}) arising from treating $p_{[n],k},\ k=1,\ldots,n$
as marginal distributions of irreversible time continuous markov
processes of pure fragmentation (or pure coagulation) on the state
space $[n]$. The study of these problems  was initiated by Kingman
and Pitman and is being extensively continued by a group of
researchers including Pitman, Bertoin, Berestycki, Gnedin et al.

\vskip .5cm In what follows we will refer to all  models induced
by the conditioning relation \refm[1] as random structures.

\section{Objective and Summary.}

 \noindent In this paper, we  study the asymptotic behaviour of the
 random
 vector
 $(K^{(n)}_1,\ldots,K^{(n)}_l)$ of small (to compare with $n$) component counts,
  defined by \refm[1], as $n\to \infty$ and $l\ge 1$ is
fixed. Based on the independence of the random variables $Z_j, \
j\ge 1$ in  \refm[1], it was a common belief that the counts
become independent, as $n\to \infty.$ We show that this is not
true in general. Our main result which is Theorem 1 in Section 3,
consists of proving that the asymptotic independence of small
component counts is equivalent to the weak convergence of the
above sequence of random vectors, and of establishing a
characterization of convergent models (multiplicative measures).
This is done by employing     the theory of $RT_\rho $ sequences
which  originated from Compton's work on logical limit laws and
has been  further developed by Burris and Bell. Based on Theorem
1, Schur's tauberian lemma and results on asymptotics of random
structures, we answer in Section 4 the question of convergence of
counting processes for the three types of basic models
distinguished in Section 1. It turns out that many models related
to multiplicative measures are divergent. In a parallel way we
discuss the problem of convergence for CFP's.  The last Section 5
contains concluding remarks, among them a historical background of
the discussed problem.
\section{Main result.}

\begin{definition}. We say that the counting process $\{{\bf K}^{(n)},\ n\ge 1\},$
 is convergent,  if for each finite \ $l\ge 1,$ the
probability law $ {\cal L}(K_1^{(n)},\ldots,K_l^{(n)})$ weakly
converges, as $n\to \infty$, to some probability law \ $F_l$  on
$\mathbb{R}^l,\ l\ge 1.$ Moreover, we say that counts
$K_1^{(n)},\ldots,K_l^{(n)}$ of small components of the random
vector\ ${\bf K}^{(n)}$ are
 asymptotically independent  if
 the above laws $F_l$ are product measures on $\mathbb{R}^l,$  for all
finite $l\ge 2.$
\end{definition}

\noindent Note that in contrast to the setting for limit shapes (see
e.g.\cite{V1}, \cite{EG1}), in this paper we study the weak
convergence of non scaled multiplicative measures.

\noindent  For a fixed $l\ge 1$,  given $k_1, \ldots, k_l$ and
sufficiently large $n$, we denote \begin{equation} M_l=\sum_{j=1}^l jk_j, \quad
and
  \quad T^{(l)}(n, M_l)=\bbb P(\sum_{j=l+1}^{n-M_l} jZ_j=n-M_l). \label{6}
\end{equation}

\noindent It is immediate that \begin{equation} T^{(l)}(n,M_l)=T^{(l)}
(n-M_l,0):=T^{(l)}_{n-M_l}, \quad l\ge 1,\quad M_l\ge 0. \label{7}
\end{equation}

\noindent Assuming  in what follows that $$a^{(j)}_0>0, \ j\ge 1,$$ we
will be dealing with the "scaled" quantities $\tilde a^{(j)}_k$,
$\tilde c_n,$ and $\tilde T^{(l)}_{n-k}$ defined  by \begin{equation} \tilde
a^{(j)}_k=\frac{ a^{(j)}_k}{ a^{(j)}_0},\quad k\ge 0,\quad j\ge 1,
\label{ak}\end{equation} \begin{equation} \tilde c_n=\big(\prod_{j=1}^n
a^{(j)}_0\big)^{-1}c_n, \quad n\ge 1, \quad \tilde c_0=1, \label{cnk}
\end{equation}\begin{equation} \tilde T^{(l)}_{n-k}= \Big(\prod_{j=l+1}^{n-k}
a^{(j)}_0\Big)^{-1}T^{(l)}_{n-k}, \quad \tilde
T^{(l)}_{0}=1.\label{11}
\end{equation}

  It will become clear from  our  discussion in Section 4 that
  in the case of classic decomposable structures,
 the  quantities $\tilde{c}_n$ and $\tilde T^{(l)}_{n-k}$
have a significant combinatorial meaning. Namely,
$p^{-n}\tilde{c}_n$ is the number of structures of size $n$ in the
case of multisets and selections, whereas  $n!\tilde{c}_n$ is the
number of assemblies of size $n$. In analogous way, $\tilde
T^{(l)}_{n-k}$ is related to the number of structures of size
$n-k$ with all  component sizes greater than $l$.

  With the help of the above  notations, we have
$$ \bbb P(
K_1^{(n)}=k_1,\ldots,K_l^{(n)}=k_l)=c_n^{-1}\big(\prod_{j=1}^l
a_{k_j}^{(j)}\big)\bbb P(\sum_{j=l+1}^n jZ_j=n-M_l)=
$$
\begin{equation} \Big(\prod_{j=1}^l \tilde a_{k_j}^{(j)}\Big) \frac{\tilde
T^{(l)}_{n-M_l}}{\tilde{ c}_n}, \label{zyz}
\end{equation}
where in the last step we have used the fact that
$$\bbb P(\sum_{j=l+1}^n
jZ_j=n-M_l)=T^{(l)}_{n-M_l}\prod_{k=1}^{M_l}\bbb P(Z_{n-M_l+k}=0)$$
and the definitions \refm[ak], \refm[cnk] and \refm[11] of the
scaled quantities.
 Note that in view of \refm[6],
$\tilde T^{(l)}_{n-M_l}$ is the same for all $k_1,\ldots, k_l:
\sum _{j=1}^l jk_j=M_l.$

 Central to our subsequent study is the notion of smoothly
growing real sequences $RT_\rho$ which definition we adopt from
\cite{Bur}, \cite{BB}.
\begin{definition}.
$RT_{\rho}, \ 0\le\rho\le\infty$ is the collection of sequences
$\{d_n\}_{n\ge 1}$ of \ nonnegative  numbers that satisfy

\begin{equation} \lim_{n\to\infty}\frac{d_n}{d_{n+1}}=\rho.   \label{9}
\end{equation}
\end{definition} \hfill \vrule height1.3ex width1.2ex depth-0.1ex

 \noindent   Sequences in $RT_\rho$ play a key role in Compton's theory of logical limit
laws  and in additive number theory (for references see
\cite{Bur}, \cite{B},\cite{BB}).

\noindent Now we are prepared to state our main result.

\begin{thm}.
\noindent  The   conditions (a),(b) and (c) below are equivalent:

\noindent(a) The counting process $\{{\bf K}^{(n)},\ n\ge 1\}$ is
convergent;

\noindent  (b) Counts of small components of random vector ${\bf
K}^{(n)}$ are asymptotically independent;

\noindent (c) $\{\tilde{ c}_n\}_{n\ge 0}\in RT_\rho,$ for some
$0\le\rho<\infty,$ and \begin{equation} 0<q^{(l)}:=\lim_{n\to
\infty}\frac{\tilde{T}_n^{(l)}}{\tilde{c}_n}=
\Big(\prod_{j=1}^l\tilde{S}^{(j)}(\rho)\Big)^{-1}<\infty, \quad
l\ge 1, \label{el}\end{equation} where
$\tilde{S}^{(j)}(\rho)=\sum_{k=0}^\infty\rho^{jk}\tilde{a}_k^{(j)}.$

\end{thm}

\noindent {\bf Proof.}

 \noindent Assumption (a) implies
that the  fraction in the RHS of \refm[zyz] has finite limits, as
$n\to \infty,$ for all fixed $M_l\ge 0.$ Moreover, for any $l\ge
1$ there exists  a $M_l\ge 0,$ such that \begin{equation} \lim_{n\to
\infty}\frac{\tilde T^{(l)}_{n-M_l}}{\tilde{c}_n}>0.\label{el1}
\end{equation}

\noindent We write now \begin{equation}\frac{\tilde
T^{(l)}_{n-M_l-1}}{\tilde{c}_n}=\left(\frac{\tilde
T^{(l)}_{n-M_{l}-1}}{\tilde{ c}_{n-1}}\right)\left(\frac{\tilde
c_{n-1}}{\tilde{c}_n}\right),\end{equation}

\noindent to conclude from the above two facts that $\tilde c_n\in
RT_\rho,$ for some $0\le\rho<\infty.$ Consequently,
  $\lim_{n\to \infty} \frac{\tilde
T^{(l)}_{n}}{\tilde c_n}:=q^{(l)}<\infty, \ l\ge 1,$ and we get
\begin{equation} \lim_{n\to \infty} \frac{\tilde T^{(l)}_{n-M_l}}{\tilde
c_n}=q^{(l)}\rho^{M_l},  \ l\ge 1,\label{bo}\end{equation} for all $
0\le M_l<\infty.$
 As a result, $0<q^{(l)}<\infty$ and
 the limit, as $n\to \infty$ of the RHS of \refm[zyz] equals to
\begin{equation} q^{(l)}\prod_{j=1}^l \tilde a_{k_j}^{(j)}\rho^{jk_j}
.\label{07}\end{equation}

\noindent \refm[07] together with \refm[zyz] show that  the
probability limit law, as $n\to\infty,$ of the vector of small
counts $(K_1^{(n)},\ldots,K_l^{(n)})$ is a product measure on
$\mathbb{R}^l,$ for all $l\ge 2.$

\noindent Thus, we proved  the implications $(a)\Rightarrow (b)$ and
$(a)\Rightarrow (c)$. Now the proof of the converse of the second
of the two implications is obvious. \hfill \vrule height1.3ex width1.2ex depth-0.1ex

\begin{remark}
(i) In  condition (c), the  case $\rho=0$ is not excluded. Setting
 $0^0=1$ and recalling that $\tilde a_{0}^{(j)}=1,\ j\ge 1,$ it follows
 from \refm[zyz] and \refm[07] that in this case the limit law of $(K_1^{(n)},\ldots,K_l^{(n)}),$  for
$l\ge 1$ is the measure concentrated at the singleton
$(0,\ldots,0)\in R^l,$ whereas $q^{(l)}=1,$  in accordance with
\refm[el], and $\lim_{n\to \infty} \frac{\tilde
T^{(l)}_{n-M_l}}{\tilde{ c}_n}=0, $ for all $M_l>0.$

\noindent (ii) Condition (c) also implies $\{\tilde T^{(l)}_{n}\}_{n\ge
0}\in RT_\rho,$ for all \ $ l\ge 1,$ with the same
$0\le\rho<\infty$ as for the sequence $\{\tilde c_n\}_{n\ge 0}.$
This can be seen by writing
$$\frac{\tilde
T^{(l)}_{n }}{\tilde{c}_n}=\left(\frac{\tilde T^{(l)}_{n}}{\tilde
T^{(l)}_{n-1}}\right)\left(\frac{\tilde
T^{(l)}_{n-1}}{\tilde{c}_n}\right)$$ and then applying the fact
that $0<q^{(l)}<\infty$ and \refm[bo].

\noindent (iii) One can see from the proof of Theorem 1 that
  condition $q^{(l)}>0, \ l\ge 1$ which is a part of the condition (c), ensures the
tightness of the corresponding sequences of finite dimensional
probability measures.
 \end{remark}

To formulate the forthcoming corollary, we need to extend the
definition \refm[ti] of the tilting transformation to the case
$\rho=0,$ in the following natural way: \begin{equation} a^{(j)}_k(0)=
  \begin{cases}
    1 , & \text{if}\ k=0 \\
    0 , & \text{otherwise}.
  \end{cases} \label{ro}
\end{equation}

\begin{cor} Let ${\bf
K}^{(n)}$ be a convergent counting process, such that  $\tilde
c_n\in RT_\rho,$ for some $ 0\le\rho<\infty.$ Then \begin{equation} \lim_{n\to
\infty}\bbb P( K_1^{(n)}=k_1,\ldots,K_l^{(n)}=k_l)=\prod_{j=1}^l
a_{k_j}^{(j)}(\rho), \quad l\ge 1, \label{111}
\end{equation}
\noindent where $ a_{k_j}^{(j)}(\rho)$  are the generic probabilities \
$ a_{k_j}^{(j)}$ tilted with the $\rho .$
\end{cor}

\noindent {\bf Proof.} By \refm[el],  \refm[07] and  the definitions
\refm[ti],\refm[ro] and  \refm[ak], it follows from Theorem 1 that
for a convergent counting process, $$ \lim_{n\to \infty}\bbb P(
K_1^{(n)}=k_1,\ldots,K_l^{(n)}=k_l)
=\prod_{j=1}^l\frac{\tilde{a}_{k_j}^{(j)}\rho^{jk}}{\tilde{S}^{(j)}(\rho)}=$$

$$
\prod_{j=1}^l\frac{a_{k_j}^{(j)}\rho^{jk}}{S^{(j)}(\rho)}=
\prod_{j=1}^l a_{k_j}^{(j)}(\rho), \ \ l\ge 1.$$
 \hfill \vrule height1.3ex width1.2ex depth-0.1ex

\begin{remark}

(i)  Corollary 1 says that asymptotic  independence with the limit
product measure composed of  generic probabilities $
a_{k_j}^{(j)}$ takes place only if $\rho=1;$

(ii)  We denote by $c_n(\theta)$ the quantity $c_n$ corresponding
to the tilting of the probabilities $a_k^{(j)}$ with $\theta\ge 0$
and recall that a multiplicative measure $\mu_n$ is invariant
under all possible tiltings  of the probabilities with $\theta>0.$
By definitions \refm[ti] and \refm[cnk] we then have
$$\widetilde{c_n(\theta)}=\theta^n\tilde{c}_n,\ n\ge 0. $$
So, if $\{\tilde{ c}_n, \ n\ge0\}\in RT_\rho,$ for some
$0\le\rho<\infty,$ then $$\{\widetilde{c_n(\theta)}, \ n\ge 0\}\in
RT_{\frac{\rho}{\theta}}.$$ This clarifies the meaning  of
Corollary 1: Given  a  convergent generic counting process such
that $\{\tilde{ c}_n, \ n\ge 0\}\in RT_\rho,$ with some $\rho>0,$
the whole family of counting processes obtained by tilting the
original one with all possible $\theta>0,$
 has the same limit
finite dimensional distributions as the counting process tilted
with the above $\rho>0$, so that the corresponding quantity
$\{\widetilde{ c_n(\rho}), \ n\ge 0\}\in RT_1.$
\end{remark}
\section{ Convergent and divergent random structures.}
We agree to call a random structure convergent (divergent) if the
corresponding to it counting process converges (diverges)
respectively, in the sense of Definition 1.

\noindent Our tool for verification   condition \refm[el] of Theorem 1
for the
 models considered, will be the remarkable Schur's tauberian lemma
cited below.
 \noindent With an obvious abuse of notation, we say that
a power series $f(x)=\sum_{n\ge 0} d_nx^n $ is in $RT_\rho$ if
$\{d_n\}_{n\ge 0}\in RT_\rho.$ We denote by * the Cauchy product,
which is a usual product extended to formal power series (see
\cite{Bur}).
\begin{lemma}(Schur (1918), see \cite{Bur},p. 62).

Let $f=f_1*f_2,$ where $f, f_1, f_2$ are power series with
 coefficients $d_n, d_n^{(1)}, d_n^{(2)}, \ n\ge 0,$
respectively,

 such that :

 (a) $f_1\in RT_\rho$ for some $0\le \rho<\infty
$ and

(b) the radius of convergence of $f_2$ is greater than $\rho.$

Then \begin{equation} \lim_{n\to \infty}\frac{ d_n}{d_n^{(1)}}=f_2(\rho).
\label{shu}\end{equation}
\end{lemma}
\vskip.5cm

We also note   the fact that $f_1\in RT_\rho$ together with
\refm[shu] imply $f\in RT_\rho.$ This can be seen by writing
$$\frac{d_{n-1}}{d_n}=\Big(\frac{d^{(1)}_{n-1}}{d^{(1)}_n}\Big)
\Big(\frac{d_{n-1}}{d_{n-1}^{(1)}}\Big)
\Big(\frac{d_{n}^{(1)}}{d_n}\Big).$$

\noindent Let $S^{(j)}$ be the generating probability function  of the
random variable $jZ_j$ in the conditioning relation \refm[1]:

 $$S^{(j)}(x)=\sum_{k\ge 0} a_{k}^{(j)} x^{kj}.$$
\noindent  Clearly, the radius of converge of $S^{(j)}$ is $\ge 1,$ for
all $j\ge 1.$

\noindent It follows from \refm[3] that the Cauchy products
$g:=\prod_{j\ge 1}S^{(j)}$ and $g_{T^{(l)}}:=\prod_{j\ge
l+1}S^{(j)}$ are the generating functions for the sequences
$\{c_n\}_{n\ge 0}$ and $\{{T}_n^{(l)}\}_{n\ge 0},\ l\ge 1,$
respectively.
 This fact is a
characteristic feature of  multiplicative measures on the set of
partitions. In accordance with the preceding notations, we also
denote by $\tilde S^{(j)},$ $\tilde g$  and $\tilde g_{\tilde
T^{(l)}}$ the generating functions for the scaled sequences
$\{\tilde a_{k}^{(j)}, \ k\ge 1\},$ $\{\tilde c_n, \ n\ge 0\}$ and
$\{\tilde T_n^{(l)}, \ n\ge 0\}$
 respectively.
 Finally, denoting  $\tilde g^{(l)}=\prod_{j=1}^l \tilde
S^{(j)},\ l\ge 1$ we have

\begin{equation} \tilde g=\tilde g_{\tilde T^{(l)}}*\tilde
g^{(l)}\label{112a}\end{equation}

\noindent or, equivalently,
 \begin{equation} \tilde g_{\tilde T^{(l)}}=\tilde
g*\left(\frac{1}{\tilde g^{(l)}}\right).\label{112}\end{equation}

Supposing that  condition (c) of Theorem 1 holds and employing
(ii) of Remark 1, allows to  apply Schur's lemma either to
\refm[112a] or to \refm[112], depending on whether $\tilde
g^{(l)}$ or $\frac{1}{\tilde g^{(l)}}$ have the radius of
convergence greater than $\rho.$ As a result we find
$\frac{1}{q^{(l)}}$ or $q^{(l)}$ respectively for the random
structures considered.

 In what follows we  examine the convergence of counting
processes for the three basic types of random  structures:
assemblies, multisets and selections, described  in Section 1.
Furthermore, the results obtained  explain the crucial difference
in the asymptotic behaviour of mean field and non mean field CFP's
associated with the above structures.

\vskip .5cm

 The implementation of Theorem 1 for random structures with given
sequences of parameters $\{m_j,\ j\ge 1\}$ requires validation of
the condition $\tilde g\in RT_\rho$ for some $0\le\rho<\infty.$
Our treatment of the problem  is based on known sufficient
conditions on  sequences $\{m_j, \ j\ge 1\}$ providing the
 above $RT_\rho$ property for  the corresponding sequences $\{\tilde{ c}_n, \ n\ge 0\}. $ The conditions  we employ are products  of two quite
different lines of research. The first is Burris-Bell theory
(\cite{B}, \cite{BB}) of $RT_\rho $ sequences, developed  with the
help of analytical tools stemming from Tauberian theory.
Motivation of this research came from
 Compton's (1980s) theory of logical limit laws and also from the additive
number system theory.
 The second source of the sufficient conditions   are
asymptotic formulae for the number of decomposable structures, in
particular recent results  by Barbour, Freiman, Stark and the
author, derived with the help of probabilistic methods. Each one
of the two approaches has its own limitations, some of which are
noted later on in this Section. In this connection, we observe
that there is little hope to obtain plausible necessary conditions
for the $ RT_\rho$ property of $\tilde g$. \\ The above mentioned
sufficient conditions combined with Theorem 1, allow us to obtain
characterizations of convergence  for each type of random
structures, and to provide examples of convergent and divergent
structures.
 \vskip.5cm

Firstly, following  \cite{bar} we call  regularly varying the
random structures   induced by random variables $Z_j, \ j\ge 1$ in
\refm[1], with \begin{equation} \bbb E Z_j\sim const \ y^j j^{\alpha}, \quad \alpha
\in \bbb R,\quad y>0,\quad j\to \infty.\label{var}\end{equation} Since
the asymptotic behaviour of regularly varying random structures
appears to be in accordance with the behaviour of the series
$\sum_{j=1}^\infty j^\alpha$ (see \cite{ABT}, \cite{frgr1},
\cite{bar}, \cite{GS}),
 it was suggested in \cite{bar} to
distinguish  the following three types of regularly varying
structures: logarithmic ($\alpha=-1$), convergent
  ($\alpha<-1$) and
expansive ($\alpha>-1$). (It goes without saying that in this
classification, the meaning of a convergent structure is different
from the one in the present paper). As in \cite{frgr1}, we extend
the above definition of expansive structures to include random
structures with $\bbb E Z_j, \ j\ge 1$ oscillating (in $j$) between
two regularly varying functions,  namely
$$(\bbb E Z_j, \ j\ge 1)\in {\cal F}(r_1,r_2;y):=\{f=f(j):  \gamma_1y^jj^{r_1-1}\le f(j) \le
\gamma_2
 y^jj^{r_2-1}, \ j\ge 1, \ y>0 \},
$$
\noindent where  $\gamma_i,\ i=1,2$ are positive constants and $0<
r_1\le r_2.$ (The  requirement $r_1,r_2>0$ is  the characteristic
feature of the expansive case).

Assuming the conditions of Schur's lemma are fulfilled, we then
get \begin{equation}
q^{(l)}=\Big(\tilde{g}^{(l)}(\rho)\Big)^{-1},\label{qls}\end{equation}
which is in accordance with \refm[el].
 \vskip.5cm

 \noindent {\bf  Assemblies} \ \ Let
$Z_j\sim Po(a_j)$, \ \  $a_j>0,\ \ j\ge 1$ . In this case,

$$\tilde a_{k_j}^{(j)}=\frac{a_j^{k_j}}{k_j!},\ \
\tilde S^{(j)}(x)=\exp{(a_jx^j)},\ \ j\ge 1$$ and $$\tilde g(x)=
\exp\left(\sum_{j\ge 1}{a_jx^j}\right),\quad \tilde{ g}^{(l)}(x)=
\exp\left(\sum_{j= 1}^l{a_jx^j}\right),\quad l\ge 1,\quad \bbb E
Z_j=a_j, \quad j\ge 1.
$$

\noindent Thus, the radius of convergence of $\frac{1}{\tilde g^{(l)}}
$ equals to $\infty$ for all finite $l\ge 1.$ Assuming $\tilde
g\in RT_\rho,\ 0\le\rho<\infty,$ we then apply  Schur's lemma to
\refm[112] with $f=\tilde g_{\tilde T^{(l)}},$ $f_1=\tilde g$ and
$f_2=\frac{1}{\tilde g^{(l)}}$ \ to get from the definition
\refm[el] that $0<q^{(l)}<\infty.$  By virtue of  Theorem 1, this
says that

\noindent \begin{proposition}  An assembly converges if and only if the
sequence $\{a_j,\ j\ge 1\}$ is such that $\tilde g\in RT_\rho,$
for some $0\le \rho<\infty.$
\end{proposition}
\noindent \begin{cor}  Assemblies with the following parameter
functions $a=\{a_j,\ j\ge 1\}$ are convergent:

(i) Smoothly growing: $a\in RT_\rho, \ 0<\rho<\infty$;

(ii) Oscillating: $a\in {\cal F}(\frac{2r}{3}+\epsilon,r;y),$
 where $r>0$, $0<\epsilon\le \frac{r}{3}$ and $y>0$.
\end{cor}
\noindent {\bf Proof} (i) follows from the important for our study
Corollary 4.3 in \cite{BB}, which says that $a\in RT_\rho, \
0<\rho<\infty$ implies $\tilde g\in RT_\rho,$ with the same
$\rho.$ For the proof of (ii), we employ the asymptotic formula
(4.99) in \cite{frgr1} which  says that for the assembly in
question with $y=1$ we have $\tilde g\in RT_1,$ and then apply
(ii) of  Remark 2 with $\theta=y$.\hfill \vrule height1.3ex width1.2ex depth-0.1ex

 Note that the
restriction on $\epsilon$ in the part (ii) of the last corollary
determines a bound on the ``size" of oscillation of $\bbb E Z_j=a_j, \
j\ge 1$ that ensures the $RT_\rho$ property for $\tilde g$. \vskip
.5cm
 \noindent \emph{Examples.} In combinatorics (see Table 2.2 in
\cite{ABT}),  many assemblies, e.g. permutations
($a_j=\frac{1}{j}$), Ewens sampling formula
($a_j=\frac{\theta}{j}, \ \theta >0$), forests of labelled rooted
trees ($a_j=\frac{j^{j-1}}{j!}\sim const\  e^jj^{-\frac{3}{2}}$),
etc are regularly varying, which implies that that in these cases
$a\in RT_{\frac{1}{y}},\ y>0,$ where $y$ is as in the definition
of a regularly varying structure. Note that in  statistical
mechanics, regularly varying assemblies with $y=1$ and $\alpha>0$
are called generalized Maxwell-Boltzman statistics (\cite{V1}.
Thus, by virtue of the condition (i) of Corollary 2, all the
 aforementioned assemblies converge.

 We give now two examples of divergent assemblies. Firstly, set
partitions ($a_j=(j!)^{-1}, \ j\ge 1$) diverges, since in this
case $\tilde g(x)=e^{e^x-1},$ so that the radius of convergence of
$\tilde g$ is infinity. In this connection note that the following
sufficient condition for $RT_\infty$ property of $\tilde g$ was
recently established in \cite{BK}:  If the parameter function $a$
is such that $gcd\{j:a_j>0\}=1$ and $a_j=O(j^{\theta j}/j!), \
0<\theta<1,\ j\to \infty,$ then $\tilde g\in RT_\infty$, which means
that the induced assembly diverges.\\
For our second example we construct an assembly with $\tilde g$
that does not belong to any class $RT_\rho, \ 0\le \rho\le
\infty.$ It is clear that the corresponding $a_j, \ j\ge 1$ should
exhibit a wild behavior. We set
$$ a_j=
  \begin{cases}
    j^{-1}, & \text{if $j\ge 1$ is odd}, \\
   j^{-1}+2^{j+1}j^{-1},  & \text{if $j\ge 2$ is even}.
  \end{cases}
$$
\noindent We then have
$$\tilde g(x)=\Big(\frac{1}{1-x}\Big)*\Big(\frac{1}{1-4x^2}\Big),$$
\noindent  which by Cauchy product formula gives

$$\tilde c_{n}=\sum_{k=0}^{[n/2]} 4^k= \frac{4^{1+[n/2]}-1}{3},\ \ n\ge 1.$$

\noindent Consequently,
$$ \lim_{n\to \infty}\frac{\tilde c_{2n-1}}{\tilde c_{2n}}=1/4$$
$$ \lim_{n\to \infty}\frac{\tilde c_{2n}}{\tilde c_{2n+1}}=1.$$

\noindent Finally, note that  for graphs on $n$ vertices,
 $\tilde {c}_n=\frac{2^{n\choose 2}}{n!}$, so that $\tilde{c}_n\in RT_0$ and  by
 (i) of Remark 1, for any $l\ge 1$
 the limit measure is  concentrated on the
singleton $(0,\ldots,0)\in \bbb R_l.$

\vskip.5cm We shift  now to considering   CFP's related to
assemblies. In this case the ratio of the net transitions
\refm[cv] has  the following form:

\begin{equation}
 q(\eta;i,j)=V(k_i,k_j,k_{i+j})
  \begin{cases}
    \frac{a_{i+j}}{a_{i}a_j}, & \text{if}\ \ i\neq j:k_ik_j>0 \\
    \frac{a_{2i}}{a_i^2} , & \text{if}\ \ i=j:k_i\ge 2,
  \end{cases}\label{ha}
\end{equation}
\noindent where \begin{equation} V(k_i,k_j,k_{i+j})=\begin{cases}
\frac{k_ik_j}{k_{i+j}+1},& \text{if}\ \ i\neq j:k_ik_j>0 \\
\frac{k_i(k_i-1)}{k_{2i}+1},& \text{if}\ \ i=j:k_i\ge 2.
\end{cases}
\end{equation}

\noindent As we explained in Section 1, such ratios correspond to mean
field CFP's with net transition rates of coagulation and
fragmentation as given by \refm[qw],\refm[143]. Consequently, the
preceding discussion reveals that amongst  mean field CFP's  both
convergent and divergent models exist. \vskip.5cm

 \noindent
{\bf Multisets} \ Assuming  $Z_j\sim NB(m_j,p^j),$ $0<p<1,$
$m_j\ge 1,$ $j\ge 1,$ we have
$$a_k^{(j)}=(1-p^j)^{m_j}\binom{m_j+k-1}{k}p^{jk},\ k\ge 0,
$$ where $p$ is a free parameter.
This gives $$ \tilde{a}_k^{(j)}=\binom{m_j+k-1}{k}p^{jk},\quad
\tilde{S}^{(j)}(x)=(1-(px)^j)^{-m_j}, \ \vert xp\vert <1,$$ which
leads to the Euler type generating function \begin{equation}\tilde
g(x)=\prod_{j\ge
1}\left(1-(px)^j\right)^{-m_j}.\label{eu}\end{equation} It is known
(see e.g. \cite{Bur}, Lemma 1.15) that $\tilde g$ converges at
some $x:\vert xp\vert <1$ if and only if
$$\lim_{j\to \infty} \left(1-(px)^j\right)^{-m_j}=1.$$ This tells
us that $\tilde g$ has the same radius of convergence $\rho$ as
the series \begin{equation}\sum_{j\ge 1} m_jp^jx^j.\label{xc}\end{equation}.

\noindent Since it is assumed that $m_j\ge 1,
 \ j\ge 1,$ we have $0\le\rho\le p^{-1}.$
 \noindent Next, in
\refm[112] the function $f_2(x)=\frac{1}{\tilde
g^{(l)}(x)}=\prod_{j=1}^{ l}\left(1-(px)^j\right)^{m_j}, \ l\ge 1$
is a polynomial, such that $f_2(p^{-1})=0,$ while $f_2(x)>0, \
0\le x< p^{-1}.$

\noindent \begin{proposition} A  multiset is convergent if and only if
$\tilde g\in RT_\rho,$ for some $\rho<p^{-1}.$
\end{proposition}
\noindent {\bf Proof.} In the case $\tilde g\in RT_{p^{-1}}$,
 we have $q ^{(l)}=0,\ l\ge 1,$ by \refm[qls] and the observation
following \refm[xc]. So, in this case a multiset diverges, because
condition (c) of Theorem 1 is violated. If now $\tilde g\in
RT_\rho$ with $0\le \rho<p^{-1},$ then $0<q_l<\infty, \ l\ge 1,$
by the same argument.\hfill \vrule height1.3ex width1.2ex depth-0.1ex

\begin{cor} Multisets with the following parameter functions
$m=\{m_j, \ j\ge 1\}$ are divergent: \\ \noindent (i) $m\in RT_1;$ (ii)
$m_j=O(j^\alpha),$ for some $\alpha\in R;$
 (iii) $m\in {\cal F}(\frac{2r}{3}+\epsilon,r;1),\ 0<\epsilon\le \frac{r}{3}, \ r>0,$

\noindent whereas multisets with  parameter functions (iv) and (v)
below converge:

\noindent (iv) $m\in RT_\rho,$ for some $0<\rho <1;$\ \ \
 (v) $m\in {\cal F}(\frac{2r}{3}+\epsilon,r;y), \ 0<\epsilon\le \frac{r}{3},
  \ y>1,\ r>0;$\\
 (vi) $m_j\asymp y^jj^{-1}, y>1. $
\end{cor}
 \noindent {\bf Proof.} Each one of the conditions (i)-(v) is
sufficient
 for $\tilde g\in RT_\rho$ with a corresponding $\rho.$ Namely, the conditions
(i) and (iv) are due to  Bell-Burris Theorems 6.1 and 6.2
(\cite{BB}), which state that  $m\in RT_\rho,$ for some $0<\rho\le
1$ implies $\tilde g(p^{-1}x)\in RT_\rho$ with the same $\rho,$
and consequently, $\tilde{g}(x)\in RT_{\rho p^{-1}}.$ (Note that
 in Theorem 6.1  in  \cite{BB},  condition (c) is  required only for
the second part of the claim).

The condition (ii) provides $\tilde g(p^{-1}x)\in RT_1,$ by the
powerful result of Bell \cite{B} that generalizes Bateman and
Erd$\ddot{o}$s theorem. So, under this condition $\tilde{g}(x)\in
RT_{ p^{-1}}.$   The conditions (iii) and (v) result from
Corollary 2 of \cite{GS} for expansive multisets, which says that
in the both cases $\tilde{g}(p^{-1}x)\in RT_{y^{-1}},$ which is
equivalent to $\tilde{g}(x)\in RT_{y^{-1}p^{-1}}$, with $y>1.$
Regarding the condition (vi), we firstly recall that $a_j\asymp
b_j$ means that the ratio ${a_j\over b_j},\ j\ge 1$ is bounded
above and below by positive constants.   The sufficiency of the
condition (vi) for $\tilde g\in RT_{y^{-1}p^{-1}}, \ y>1$ was
proven in \cite{ST} devoted to logical limit laws for logarithmic
multisets. \hfill \vrule height1.3ex width1.2ex depth-0.1ex

\vskip.5cm
 \emph{Examples.} Integer partitions
($m_j=1, \ j\ge 1),$  planar partitions ($ m_j=j, \ j\ge 1$) (see
\cite{A}) and generalized Bose-Einstein statistics ($m_j=j^\alpha,
\ \alpha>0$) diverge, since in all these cases $m\in RT_1.$ The
following logarithmic multisets (see \cite{ABT}) with $m\in
RT_\rho, \ \rho<1$ converge : mapping patterns
($m_j=\frac{b^{-j}}{2j}, \ b<1,\ j\ge 1),$ monic polynomials over
$GF(q), \ q>1$ ($m_j\sim \frac{q^j}{j}).$ Also converge  forests
of unlabeled unrooted ($m_j\sim const \ b^{-j}j^{-5/2}, \ b<1)$
and rooted ($m_j\sim const \
 b^{-j}j^{-3/2},\ b<1)$ trees.
\vskip.5cm \begin{remark}  There is a formal  linkage between
assemblies and multisets,  expressed as follows. We saw that in
the case of assemblies the generating fuction $\tilde g$ is of the
exponential form: $\tilde g(x)=e^{Q(x)},$ where $Q(x)=\sum_{j\ge
1}a_j x^j,$ with $a_j\ge 0,\ j\ge 1.$  For multisets, the Euler
type generating function $\tilde g$ in \refm[eu], can  be written
in the same form, with the function $Q^*$called the star
transformation (see \cite{B}, \cite{BB}) of the generic generating
function $Q(x)=\sum_{j\ge 1}m_jp^jx^j$ of the sequence $m:$
$$Q^*(x)=\sum_{j\ge 1}m^*_jx^j,\quad
m^*_j=\sum_{lk=j}\frac{m_l p^l}{k}, \quad j\ge 1, \quad m^*_0=0.$$
Thus, $m^*_j\ge 0, \ j\ge 1,$ which is a basic assumption in the
theory of $RT_\rho$ sequences. It was proven in \cite{BB} that if
$m_jp^j\in RT_\rho,$ with some $0<\rho<1,$ then $m_j^*\sim
m_jp^j,\
 j\to \infty,$ which means that in this case  the  multiset
 behaves asymptotically as the assembly induced by
$Z_j\sim Po(m_jp^j), \ j\ge 1.$ This fact explains the condition
 (iv) of Corollary 3.

\end{remark}

Regarding the associated CFP's, for multisets
 \refm[cv] becomes:

\begin{equation} q(\eta;i,j)=V(k_i,k_j,k_{i+j})
  \begin{cases}
   \frac{m_{i+j}+k_{i+j}}{(m_i+k_i-1)(m_j+k_j-1)} , & \text{if}\ \ i\neq j:k_ik_j>0 \\
  \frac{m_{2i}+k_{2i}}{(m_i+k_i-1)(m_i+k_i-2)} , & \text{if}\ \ i=j:k_i\ge
  2,
  \end{cases}\label{hav}
\end{equation}

\noindent where $V(k_i,k_j,k_{i+j})$ is as in \refm[ha].
 Since  the second factor in \refm[hav] depends both on
$\eta=(k_1,\ldots,k_n)$ and the parameters $m_j,\ j\ge 1$ of the
CFP, the process is not a mean field model. As we mentioned in
section 1, this fact is especially clear in the case $m_j=1,\ j\ge
1$ that corresponds to the uniform measure $\mu_n$ on the set
$\Omega_n,$ so that we have from \refm[hav]
$$ q(\eta;i,j)=1,\ \eta\in \Omega_n,\ i,j\ge 1,\ i+j\le n. $$

\noindent {\bf Selections.} In this case  $Z_j\sim
Bi(\frac{p^j}{1+p^j};m_j),\ m_j\ge 1,\ j\ge 1$ and $p>0.$
 Hence, $$\tilde g(x)=\prod_{j\ge
 1}\left(1+(px)^j\right)^{m_j}.$$ So, as in the case of multisets,
 $\tilde g$
has the same radius of convergence $\rho\le p^{-1}$ as the series
\refm[xc],
 while the radius
of convergence of $\tilde g^{(l)}$  equals  $\infty$ for all $l\ge
1.$

\noindent \begin{proposition} A selection is convergent if and only if
the sequence $\{m_j\}_{j\ge 1}$ is such that $\tilde g\in
RT_\rho,$ for some $0\le \rho\le p^{-1}.$
\end{proposition}
\noindent {\bf Proof:} Since the radius of convergence of
$\frac{1}{\tilde g^{(l)}}$ is $ p^{-1},$ for all $l\ge 1,$ the
preceding discussion implies that Schur's lemma can not be applied
to \refm[112]. So, we apply the lemma to  \refm[112a] to get
$$\infty> q^{(l)}=\left(\tilde g^{(l)}(\rho)\right)^{-1}>0,\ \
l\ge 1,\quad 0\le \rho\le p^{-1}.
$$ Now the claim follows
  from Theorem 1.\hfill \vrule height1.3ex width1.2ex depth-0.1ex

\noindent Comparing Propositions 3 and 4, one sees that for selections
the conditions of convergence are less strict than the ones for
multisets. \vskip.5cm

 For selections,  conditions on $m_j, \ j\ge 1$ of
Bell-Burris type providing $\tilde g\in RT_\rho$ are not known.
This is due to  the fact that for selections  the star
transformation does not ensure the nonnegativity of  $m_j^*, \
j\ge 1.$

However, the  probabilistic method for enumeration of decomposable
structures works in the case of selections either. It follows from
Theorem 5 in \cite{GS} that for expansive selections, $m\in
{\cal F}(\frac{2r}{3}+\epsilon,r;y), \ r,y>0,$ implies that $\tilde
g\in RT_\rho,$ with $\rho=(yp)^{-1}.$ By Proposition 4, this
provides convergence of the corresponding selections, if $y\ge 1$.
As a result, we derive the convergence of the following expansive
selections: integer partitions into distinct parts ($m_j=1,\ j\ge
1$) and generalized Fermi statistics ($m_j=j^\alpha,\ \alpha>0).$

For the associated CFP's we obtain from \refm[cv]  \begin{equation}
q(\eta;i,j)=V(k_i,k_j,k_{i+j})
  \begin{cases}
   \frac{m_{i+j}-k_{i+j}+1}{(m_i-k_i-1)(m_j-k_j-1)} , & \text{if}\ \ i\neq j:k_ik_j>0,\
   0\le k_i\le m_i \\
  \frac{m_{2i}-k_{2i}}{(m_i-k_i+1)(m_i-k_i+2)} , & \text{if}\ \ i=j:2\le
  k_i\le m_i.
  \end{cases}\label{hav1}
\end{equation}

 This shows that these CFP's are not mean field models.

\section{ Concluding remarks and history}

Generally speaking, the phenomenon of asymptotic independence  of
a finite number of small groups of particles  in large random
systems (i.e. systems formed of  a large number of randomly
interacting particles) was observed in different fields of
applications, under various mathematical settings. The assumption
of asymptotic independence, sometimes accepted without proof, was
of great help for the study of probabilistic models considered.
Without surveying the subject, we point out below a few settings
parallel (in some sense) to the one in the present paper.

(i) {\bf Mean field models.} In statistical physics, there is a
general belief that when the range of interaction between
individual groups of particles is large enough, the aforementioned
phenomenon of asymptotic independence occurs. The assumption of
independence is  the main idea behind the Ginzburg-Landau
equations and it is the basis of
 calculations in a
variety of mean field models and models that can be approximated
by a mean field (see e.g. \cite{dur}, p. 111-113). Recall that in
mean field models, each group of particles is allowed to interact
with all other ones, so that the topology of a state space does
not play a role. An example of a mean field is an interacting spin
system  on a complete graph with $n$ vertices. Such a system is
given by a time continuous Markov process on a state space
$\{0,1\}^n$, with given infinitesimal (in time) rates of flips
$0\rightarrow 1$ and $1\rightarrow 0$ (for references on
interacting particle systems see \cite{lig}). It is natural to
suggest that under
 some conditions  on the rates of flips, the equilibrium states of
  a given finite set of
vertices of a complete graph become asymptotically independent, as
$n\to \infty.$ For a spin system called
 the noisy voter model the above fact was proven in \cite{GM} for pairs of states.
(The pairwise independence of states at equilibrium  follows from
formula  for covariance in Example 1, p.38 in \cite{GM}, after
adding the missing factor $r=N-1$ in the exponent of the integrand
and passing to the limit as $N,t\to \infty$). A good discussion of
mean field approximations for systems of interacting particle is
presented in \cite{BY}.\\ From the above description, it is clear
that in contrast to the setting \refm[1] in the present paper, the
formulation of a mean field model does not assume any
conditioning.

While mean field theory arose  in statistical physics it has been
recently applied elsewhere, for example in neural networks, fluid
mechanics and artificial intelligence, where it is used as an
approximation of complex (in respect of interactions) many-body
systems in equilibrium. The main idea of the mean field
approximation is to replace all interactions to any one body with
one average interaction. As a result, the mean field partition
function associated with a hamiltonian of a system considered
becomes a product of the mean field local partition functions,
which greatly simplifies the calculations (for references see e.g.
Section 6.1.4 in \cite{Li}).

\vskip.5cm

\noindent (ii){\bf The Gibbs conditioning principle} (see \cite{DF},
\cite{DZ}, \cite{GZ}). In the context of an ideal gas model,  the
simplest version of the principle reads as follows. Let $X_1,
X_2,\ldots$ be independent and identically distributed random
variables viewed as energies of individual particles, so that
$X_1+\ldots X_n$ is the total energy of a system of $n$ particles.
Let $\bbb E X_1=1$ and assume some suitable regularity conditions on a
common probability law $P$ of the sequence of random variables.
Then, for a fixed $k\ge 1$ the distribution law $\Lambda_{k,n}$ of
$(X_1,\ldots, X_k)$ conditioned on $X_1+\ldots X_n=n$ weakly
converges, as $n\to \infty$ to the $k-$ fold product law $P^k.$ In
statistical physics the law $\Lambda_{k,n}$ is called a
microcanonical distribution, and the gibbs conditional principle
asserts asymptotic independence of energies  of any finite number
of particles in microcanonical ensembles. Formulated in the
beginning of the $20$-th century, the principle has been extended
and refined in different directions, with particular attention
being paid to the rate of convergence to  limit distributions. \\
To distinguish from the conditioning relation \refm[1],  the
measure $\Lambda_{k,n}$ is defined on the simplex, from which it
follows that the Cauchy product relationship \refm[112a], which is
basic for the study of multiplicative measures $\mu_n$, is not
valid. However, we believe that the interplay between the above
two settings deserves further study. Quite independently, the
distribution $\Lambda_{k,n},$ with $k=n$ and discrete and not
necessary identically distributed random variables $X_1,X_2,
\ldots$ was introduced by Kolchin (\cite{Kol}) for  representation
of distributions of cell counts in combinatorial urn schemes. In
\cite{Kol} the representation is called the generalized scheme of
allocation, whereas in \cite{BP} and \cite{P}, it is named Kolchin
representation formula. We note that the problem of asymptotic
independence is not addressed in \cite{Kol}.

 \noindent (iii) {\bf Random combinatorial structures}.
In the theory of random structures, the asymptotic independence of
numbers of small groups was discussed in numerous papers, starting
from the 1940's. A general set up leading to asymptotic estimation
of the total variation distance between component spectrum of
small counts (as defined by the conditioning relation \refm[1])
and the independent process was developed by Arratia and
Tavar$\acute{e}$ in their seminal paper \cite{AT} (see also
\cite{ABT},\cite{fri} and \cite{Kol}). As a result, asymptotic
independence was established for  logarithmic random structures
with $y=1$.
In \cite{bar}, the same was proven for convergent structures, and
in \cite{frgr1}, the asymptotic independence was proven  for
expansive assemblies with any $y>0$. Regarding assemblies,
multisets and expansive selections,  the aforementioned results
easily follow from our results in Section 4. In fact, recall that
for assemblies $\bbb E Z_j=a_j,\ j\ge 1,$,  for multisets $\bbb E
Z_j=\frac{m_jp^j}{1-p^j}, \ j\ge 1,\ 0<p<1$ and for selections $\bbb E
Z_j=\frac{m_jp^j}{1+p^j}, \ j\ge 1, \ 0<p<1.$ By the definition
\refm[var] of a regularly varying structure, we thereby conclude
that the following facts hold.
\begin{itemize} \item For assemblies,  $a=\{a_j, \ j\ge 1\}\in
RT_{y^{-1}}, \ y>0,$ for all $\alpha\in \bbb R,$ which implies
convergence by the condition (i) of Corollary 2.

\item For multisets, $m=\{m_j,  \ j\ge 1\}\in RT_{y^{-1}p}, \
0<p<1,$ for all $\alpha\in R.$ This implies   convergence if $y\ge
1,$ by the condition (iv) of Corollary 3. Note that in the case
$y=p,$ a regularly varying multiset diverges for all $\alpha\in
\bbb R,$ by the condition (i) of Corollary 3.

\item For selections, $m_j\sim const j^\alpha (yp^{-1})^j, \ j\ge
1.$ By the discussion following Proposition 4, this provides
convergence for all $\alpha>0$ and     $y\ge 1.$

\end{itemize}

 In this connection, we mention
 that to our knowledge no examples of random
structures for which the independence principles fails were given
in the literature, prior to this paper.

 Our results reveal also a basic difference
between  pictures of asymptotics clustering of components in
general convergent structures and regularly varying divergent
multisets. Namely, in the case of a convergent structure,
Corollary 1 tells us that with a positive limit probability there
are components of any fixed sizes, i.e.
$$\lim_{n\to\infty}\bbb P(K^{(n)}_1=k_1,\ldots,K^{(n)}_l=k_l)>0,\quad
k_1,\ldots,k_l\ge 0,\quad l\ge 1.$$ On the other hand, if a
regularly varying multiset diverges, then  $  q^{(l)}=0, \ l\ge
1,$ by Proposition 3, from which it follows that
$\lim_{n\to\infty}\bbb P(K^{(n)}_j=k_j)=0,$ for all finite $k_j\ge 0,
\ j\ge 1.$

\vskip.5cm (iv) {\bf CFP's}. It is common to trace the beginning
of rigorous mathematical models of coagulation-fragmentation to
the paper by Smoluchowski (1918) where the famous system of
coagulation equations describing the time evolution of the process
was derived. Already in this paper the assumption of independence
(more precisely, the absence of correlations)  of clusters of
small sizes was adopted. Subsequently,  deterministic and
stochastic versions of the model were studied in numerous papers
in probability and various applied fields. The study of reversible
CFP's was concentrated on what we call in the present paper mean
field CFP's (\cite{W}, \cite{K}). (Recall that these models
conform to assemblies). In \cite{DGG} the model was treated as a
reversible Markov chain on the set of partitions and it was proven
(Theorem 4, (4.24)) that, if  $\tilde g\in RT_\rho, \
0\le\rho<\infty,$ then at the equilibrium of the process,
$$cov(K_l^{(n)},K_m^{(n)})\to 0,  \ n\to \infty,$$ for any fixed
$l\neq m.$ This is, of course, a weak form of our Proposition 2.
More details on the history of CFP's can be found in \cite{frgr1}.

 \vskip.5cm
 {\bf Acknowledgement.} This research was supported
by New York Metropolitan Research Fund. A part of the present
paper was written during my visit to the School of Mathematical
Sciences, at Monash University. I am grateful to Fima, Klebaner,
Aiden Sudbury and Kais Hamza for their hospitality. I also thank a
referee for  some very helpful suggestions.

redpajama_set_name: RedPajamaArXiv