Probability distribution
In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn (Chinese: 藺鴻圖) in his 2017 study of daily distributions of the S&amp;P 500 and the VIX. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.
Of the three parameters defining the distribution, the stability parameter formula_1 is most important. Stable count distributions have formula_2. The known analytical case of formula_3 is related to the VIX distribution (See Section 7 of ). All the moments are finite for the distribution.
Definition.
Its standard distribution is defined as
 formula_4
where formula_5 and formula_6
Its location-scale family is defined as
 formula_7
where formula_8, formula_9, and formula_6
In the above expression, formula_11 is a one-sided stable distribution, which is defined as following.
Let formula_12 be a standard stable random variable whose distribution is characterized by formula_13, then we have
 formula_14
where formula_2.
Consider the Lévy sum formula_16 where formula_17, then formula_18 has the density formula_19 where formula_20. Set formula_21, we arrive at formula_22 without the normalization constant.
The reason why this distribution is called "stable count" can be understood by the relation formula_23. Note that formula_24 is the "count" of the Lévy sum. Given a fixed formula_1, this distribution gives the probability of taking formula_24 steps to travel one unit of distance.
Integral form.
Based on the integral form of formula_11 and formula_28, we have the integral form of formula_29 as
formula_30
Based on the double-sine integral above, it leads to the integral form of the standard CDF:
formula_31
where formula_32 is the sine integral function.
The Wright representation.
In "Series representation", it is shown that the stable count distribution is a special case of the Wright function (See Section 4 of ):
formula_33
This leads to the Hankel integral: (based on (1.4.3) of )
formula_34where Ha represents a Hankel contour.
Alternative derivation – lambda decomposition.
Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of )
 formula_35where formula_2.
Let formula_37, and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,
formula_38
where formula_39.
This is called the "lambda decomposition" (See Section 4 of ) since the LHS was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as "exponential power distribution", or the "generalized error/normal distribution", often referred to when formula_40. It is also the Weibull survival function in Reliability engineering.
Lambda decomposition is the foundation of Lihn's framework of asset returns under the stable law. The LHS is the distribution of asset returns. On the RHS, the Laplace distribution represents the lepkurtotic noise, and the stable count distribution represents the volatility.
Stable Vol distribution.
A variant of the stable count distribution is called the stable vol distribution formula_41. 
The Laplace transform of formula_42 can be re-expressed in terms of a Gaussian mixture of formula_41 (See Section 6 of ).
It is derived from the lambda decomposition above by a change of variable such that
formula_44
where
formula_45
This transformation is named generalized Gauss transmutation since it generalizes the Gauss-Laplace transmutation, which is equivalent to formula_46.
Connection to Gamma and Poisson distributions.
The shape parameter of the Gamma and Poisson Distributions is connected to the inverse of Lévy's stability parameter formula_47. 
The upper regularized gamma function formula_48 can be expressed as an incomplete integral of formula_49 as 
formula_50
By replacing formula_49 with the decomposition and carrying out one integral, we have:
formula_52
Reverting formula_53 back to formula_54, we arrive at the decomposition of formula_48 in terms of a stable count: 
formula_56
Differentiate formula_48 by formula_58, we arrive at the desired formula:
formula_59
This is in the form of a product distribution. The term formula_60 in the RHS is associated with a Weibull distribution of shape formula_61. Hence, this formula connects the stable count distribution to the probability density function of a Gamma distribution (here) and the probability mass function of a Poisson distribution (here, formula_62). And the shape parameter formula_61 can be regarded as inverse of Lévy's stability parameter formula_47.
Connection to Chi and Chi-squared distributions.
The degrees of freedom formula_65 in the chi and chi-squared Distributions can be shown to be related to formula_66. Hence, the original idea of viewing formula_67 as an integer index in the lambda decomposition is justified here.
For the chi-squared distribution, it is straightforward since the chi-squared distribution is a special case of the gamma distribution, in that formula_68. And from above, the shape parameter of a gamma distribution is formula_47.
For the chi distribution, we begin with its CDF formula_70, where formula_71. Differentiate formula_70 by formula_58 , we have its density function as
 formula_74
This formula connects formula_75 with formula_1 through the formula_77 term.
Connection to generalized Gamma distributions.
The generalized gamma distribution is a probability distribution with two shape parameters, and is the super set of the gamma distribution, the Weibull distribution, the exponential distribution, and the half-normal distribution. Its CDF is in the form of formula_78.
Differentiate formula_82 by formula_58, we arrive at the product-distribution formula:
formula_84
where formula_85 denotes the PDF of a generalized gamma distribution, 
whose CDF is parametrized as formula_82. 
This formula connects formula_87 with formula_1 through the formula_89 term. The formula_90 term is an exponent representing the second degree of freedom in the shape-parameter space.
This formula is singular for the case of a Weibull distribution since formula_61 must be one for formula_92;
but for formula_93 to exist, formula_61 must be greater than one. 
When formula_95, formula_93 is a delta function and this formula becomes trivial.
The Weibull distribution has its distinct way of decomposition as following.
Connection to Weibull distribution.
For a Weibull distribution whose CDF is formula_97, its shape parameter formula_98 is equivalent to Lévy's stability parameter formula_99. 
A similar expression of product distribution can be derived, such that the kernel is either 
a one-sided Laplace distribution formula_100 or a Rayleigh distribution formula_101. 
It begins with the complementary CDF, which comes from Lambda decomposition:
formula_102
By taking derivative on formula_58, we obtain the product distribution form of a Weibull distribution PDF formula_104 as
formula_105
where formula_106 and formula_107.
it is clear that formula_108 from the formula_109 and formula_110 terms.
Asymptotic properties.
For stable distribution family, it is essential to understand its asymptotic behaviors. From, for small formula_111,
formula_112
This confirms formula_113.
For large formula_111,
formula_115
This shows that the tail of formula_116 decays exponentially at infinity. The larger formula_117 is, the stronger the decay.
This tail is in the form of a generalized gamma distribution, where in its formula_118 parametrization,
formula_119, formula_120, and formula_121. 
Hence, it is equivalent to formula_122, 
whose CDF is parametrized as formula_123.
Moments.
The "n"-th moment formula_124 of formula_22 is the formula_126-th moment of formula_11. All positive moments are finite. This in a way solves the thorny issue of diverging moments in the stable distribution. (See Section 2.4 of )
formula_128
The analytic solution of moments is obtained through the Wright function:
formula_129
where formula_130(See (1.4.28) of )
Thus, the mean of formula_22 is
formula_132
The variance is
formula_133
And the lowest moment is formula_134 by applying 
formula_135 when formula_136.
The "n"-th moment of the stable vol distribution formula_137 is
formula_138
Moment generating function.
The MGF can be expressed by a Fox-Wright function or Fox H-function:
formula_139
As a verification, at formula_140, formula_141 (see below) can be Taylor-expanded to formula_142 via formula_143.
Known analytical case – quartic stable count.
When formula_144, formula_145 is the Lévy distribution which is an inverse gamma distribution. Thus formula_146 is a shifted gamma distribution of shape 3/2 and scale formula_147,
 formula_148
where formula_8, formula_9.
Its mean is formula_151 and its standard deviation is formula_152. This called "quartic stable count distribution". The word "quartic" comes from Lihn's former work on the lambda distribution where formula_153. At this setting, many facets of stable count distribution have elegant analytical solutions.
The "p"-th central moments are formula_154. The CDF is formula_155 where formula_156 is the lower incomplete gamma function. And the MGF is formula_157. (See Section 3 of )
Special case when α → 1.
As formula_158 becomes larger, the peak of the distribution becomes sharper. A special case of formula_22 is when formula_160. The distribution behaves like a Dirac delta function,
formula_161
where formula_162, and formula_163.
Likewise, the stable vol distribution at formula_164 also becomes a delta function,
formula_165
Series representation.
Based on the series representation of the one-sided stable distribution, we have:
formula_166.
This series representation has two interpretations:
formula_171
The proof is obtained by the reflection formula of the Gamma function: formula_172, which admits the mapping: formula_173 in formula_170. The Wright representation leads to analytical solutions for many statistical properties of the stable count distribution and establish another connection to fractional calculus.
Applications.
Stable count distribution can represent the daily distribution of VIX quite well. It is hypothesized that VIX is distributed like formula_175 with formula_176 and formula_177 (See Section 7 of ). Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, formula_178 is called the "floor volatility". In practice, VIX rarely drops below 10. This phenomenon justifies the concept of "floor volatility". A sample of the fit is shown below: 
One form of mean-reverting SDE for formula_179 is based on a modified Cox–Ingersoll–Ross (CIR) model. Assume formula_180 is the volatility process, we have
formula_181
where formula_182 is the so-called "vol of vol". The "vol of vol" for VIX is called VVIX, which has a typical value of about 85.
This SDE is analytically tractable and satisfies the Feller condition, thus formula_180 would never go below formula_178. But there is a subtle issue between theory and practice. There has been about 0.6% probability that VIX did go below formula_178. This is called "spillover". To address it, one can replace the square root term with formula_186, where formula_187 provides a small leakage channel for formula_180 to drift slightly below formula_178.
Extremely low VIX reading indicates a very complacent market. Thus the spillover condition, formula_190, carries a certain significance - When it occurs, it usually indicates the calm before the storm in the business cycle.
Generation of Random Variables.
As the modified CIR model above shows, it takes another input parameter formula_191 to simulate sequences of stable count random variables. The mean-reverting stochastic process takes the form of
 formula_192
which should produce formula_193 that distributes like formula_194 as formula_195. 
And formula_191 is a user-specified preference for how fast formula_197 should change. 
By solving the Fokker-Planck equation, the solution for formula_198 in terms of formula_199 is
 formula_200
It can also be written as a ratio of two Wright functions, 
 formula_201
When formula_202, this process is reduced to the modified CIR model where
formula_203. 
This is the only special case where formula_204 is a straight line.
Likewise, if the asymptotic distribution is formula_41 as formula_195,
the formula_204 solution, denoted as formula_208 below, is
 formula_209
When formula_210, it is reduced to a quadratic polynomial:
formula_211. 
Stable Extension of the CIR Model.
By relaxing the rigid relation between the formula_212 term and the formula_191 term above,
the stable extension of the CIR model can be constructed as
 formula_214
which is reduced to the original CIR model
at formula_202: 
formula_216. 
Hence, the parameter formula_80 controls the mean-reverting speed, 
the location parameter formula_218 sets where the mean is, formula_191 is the volatility parameter, 
and formula_1 is the shape parameter for the stable law.
By solving the Fokker-Planck equation, the solution for the PDF formula_221 at formula_222 is
 formula_223
To make sense of this solution, consider asymptotically for large formula_58, formula_221's tail is still in the form of a generalized gamma distribution, where in its formula_226 parametrization,
formula_119, 
formula_228, and 
formula_229. 
It is reduced to the original CIR model
at formula_202 where formula_231 
with formula_232 and 
formula_233; hence
formula_234.
Fractional calculus.
Relation to Mittag-Leffler function.
From Section 4 of, the inverse Laplace transform formula_235 of the Mittag-Leffler function formula_236 is (formula_237)
formula_238
On the other hand, the following relation was given by Pollard (1948),
formula_239
Thus by formula_240, we obtain the relation between stable count distribution and Mittag-Leffter function:
formula_241
This relation can be verified quickly at formula_144 where formula_243 and formula_244. This leads to the well-known quartic stable count result:
formula_245
Relation to time-fractional Fokker-Planck equation.
The ordinary Fokker-Planck equation (FPE) is formula_246, where formula_247 is the Fokker-Planck space operator, formula_248 is the diffusion coefficient, formula_249 is the temperature, and formula_250 is the external field. The time-fractional FPE introduces the additional fractional derivative formula_251 such that formula_252, where formula_253 is the fractional diffusion coefficient.
Let formula_254 in formula_235, we obtain the kernel for the time-fractional FPE (Eq (16) of )
formula_256
from which the fractional density formula_257 can be calculated from an ordinary solution formula_258 via
formula_259
Since formula_260 via change of variable formula_261, the above integral becomes the product distribution with formula_262, similar to the "lambda decomposition" concept, and scaling of time formula_263:
formula_264
Here formula_265 is interpreted as the distribution of impurity, expressed in the unit of formula_266, that causes the anomalous