Discrete-stable distribution explained

The discrete-stable distributions^[1] are a class of probability distributions with the property that the sum of several random variables from such a distribution under appropriate scaling is distributed according to the same family. They are the discrete analogue of the continuous-stable distributions.

The discrete-stable distributions have been used in numerous fields, in particular in scale-free networks such as the internet, social networks^[2] or even semantic networks.^[3]

Both the discrete and continuous classes of stable distribution have properties such as infinite divisibility, power law tails and unimodality.

The most well-known discrete stable distribution is the Poisson distribution which is a special case.^[4] It is the only discrete-stable distribution for which the mean and all higher-order moments are finite.

Definition

The discrete-stable distributions are defined^[5] through their probability-generating function

G(s| \nu,a) = \sum ^{inf ty} _{n =0} P(N| \nu,a)( 1- s) ^{N} = \exp( - a s ^{\nu).}

In the above,

a> 0 is a scale parameter and 0 <\nu\le 1 describes the power-law behaviour such that when 0 <\nu< 1 ,\lim _{N} P(N|\nu,a) \sim (1)/(N ^{\nu+1}) .

When

\nu =1 the distribution becomes the familiar Poisson distribution with mean a .

The characteristic function of a discrete-stable distribution has the form:^[6]

\varphi(t; a, \nu) = \exp \left[a \left( e ^{it} - 1 \right) ^{\nu} \right] , with a> 0 and 0 <\nu\le 1 .

Again, when

\nu =1 the distribution becomes the Poisson distribution with mean a .

The original distribution is recovered through repeated differentiation of the generating function:

P(N|\nu,a) = \left. (( -1 ) ^{N})/(N!) (d ^{NG(s|\nu,a)})/(ds ^{N}) \right| _{s=1} .

A closed-form expression using elementary functions for the probability distribution of the discrete-stable distributions is not known except for in the Poisson case, in which

P(N| \nu =1 , a) = (a ^{N} e ^{-a})/(N!) .

Expressions do exist, however, using special functions for the case

\nu =1 / 2 ^[7] (in terms of Bessel functions) and \nu =1 / 3 ^[8] (in terms of hypergeometric functions).

As compound probability distributions

The entire class of discrete-stable distributions can be formed as Poisson compound probability distributions where the mean,

λ , of a Poisson distribution is defined as a random variable with a probability density function (PDF). When the PDF of the mean is a one-sided continuous-stable distribution with stability parameter 0 < \alpha < 1 and scale parameter c the resultant distribution is discrete-stable with index \nu = \alpha and scale parameter a = c \sec( \alpha \pi / 2 ) .

Formally, this is written:

P(N| \alpha, c \sec( \alpha \pi / 2 )) = \int ^{inf ty} _{0} P(N| 1 , λ )p( λ ; \alpha, 1 , c, 0 ) d λ

where

p(x; \alpha, 1 , c, 0 ) is the pdf of a one-sided continuous-stable distribution with symmetry paramètre \beta =1 and location parameter \mu = 0 .

A more general result states that forming a compound distribution from any discrete-stable distribution with index

\nu with a one-sided continuous-stable distribution with index \alpha results in a discrete-stable distribution with index \nu ⋅ \alpha , reducing the power-law index of the original distribution by a factor of \alpha .

In other words,

P(N| \nu ⋅ \alpha, c \sec(\pi \alpha / 2 )) = \int ^{inf ty} _{0} P(N| \alpha, λ )p( λ ; \nu, 1 , c, 0 ) d λ .

In the Poisson limit

In the limit

\nu \rarr 1 , the discrete-stable distributions behave^[9] like a Poisson distribution with mean a \sec(\nu \pi / 2 ) for small N , however for N \gg 1 , the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails

P(N) \sim 1 /N ^{1} to a discrete-stable distribution is extraordinarily slow^[10] when \nu ≈ 1 - the limit being the Poisson distribution when \nu > 1 and P(N| \nu, a) when \nu \leq 1 .

See also

• Stable distribution
• Poisson distribution

Further reading

• Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley.
• Book: Gnedenko . B. V. . Kolmogorov . A. N. . 1954. Limit Distributions for Sums of Independent Random Variables. registration . Addison-Wesley.
• Book: Ibragimov . I. . Linnik . Yu . 1971. Independent and Stationary Sequences of Random Variables . Wolters-Noordhoff Publishing Groningen, The Netherlands .

Notes and References

1. Steutel, F. W.. van Harn, K.. Discrete Analogues of Self-Decomposability and Stability. Annals of Probability. 1979. 7. 5. 893–899. 10.1214/aop/1176994950. free.
2. Barabási, Albert-László (2003). Linked: how everything is connected to everything else and what it means for business, science, and everyday life. New York, NY: Plum.
3. Steyvers, M.. Tenenbaum, J. B.. The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth. Cognitive Science. 2005. 29. 1. 41–78. 10.1207/s15516709cog2901_3. 21702767. cond-mat/0110012. 6000627.
4. Book: Renshaw, Eric . Stochastic Population Processes: Analysis, Approximations, Simulations . 2015-03-19 . OUP Oxford . 978-0-19-106039-7 . en.
5. Hopcraft, K. I.. Jakeman, E. . Matthews, J. O. . Generation and monitoring of a discrete stable random process. Journal of Physics A. 2002. 35. 49. L745–752. 10.1088/0305-4470/35/49/101. 2002JPhA...35L.745H.
6. Web site: Modeling financial returns by discrete stable distributions. Slamova, Lenka. Klebanov, Lev. International Conference Mathematical Methods in Economics. 2023-07-07.
7. Matthews, J. O.. Hopcraft, K. I. . Jakeman, E. . Generation and monitoring of discrete stable random processes using multiple immigration population models. Journal of Physics A. 2003. 36. 46 . 11585–11603. 10.1088/0305-4470/36/46/004. 2003JPhA...3611585M.
8. PhD . Continuous and discrete properties of stochastic processes . Lee . W.H. . 2010 . The University of Nottingham .
9. Lee, W. H.. Hopcraft, K. I. . Jakeman, E. . Continuous and discrete stable processes. Physical Review E. 2008. 77. 1. 011109–1 to 011109–04. 10.1103/PhysRevE.77.011109. 18351820 . 2008PhRvE..77a1109L.
10. Hopcraft, K. I.. Jakeman, E. . Matthews, J. O. . Discrete scale-free distributions and associated limit theorems. Journal of Physics A. 2004. 37. 48. L635–L642. 10.1088/0305-4470/37/48/L01. 2004JPhA...37L.635H.