Discrete-stable distribution

Discrete-stable distributions{{cite journal|last=Steutel, F. W.|author2=van Harn, K.|title=Discrete Analogues of Self-Decomposability and Stability|journal=Annals of Probability|date=1979|volume=7|issue=5|pages=893–899|doi=10.1214/aop/1176994950|url=https://pure.tue.nl/ws/files/1956807/Metis199408.pdf|doi-access=free}} 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 continuous-stable distributions.

Discrete-stable distributions have been used in numerous fields, in particular in scale-free networks such as the internet and social networksBarabási, 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. or even semantic networks.{{cite journal|last=Steyvers, M.|author2=Tenenbaum, J. B.|title=The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth|journal=Cognitive Science|date=2005|volume=29|issue=1|pages=41–78|doi=10.1207/s15516709cog2901_3|pmid=21702767|arxiv=cond-mat/0110012|s2cid=6000627}}

Both 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 special case of the Poisson distribution.{{Cite book |last=Renshaw |first=Eric |url=https://books.google.com/books?id=pqE1CgAAQBAJ&pg=PA134 |title=Stochastic Population Processes: Analysis, Approximations, Simulations |date=2015-03-19 |publisher=OUP Oxford |isbn=978-0-19-106039-7 |language=en}} It is the only discrete-stable distribution for which the mean and all higher-order moments are finite.{{dubious|date=December 2016}}

Definition

The discrete-stable distributions are defined{{cite journal|last=Hopcraft, K. I.|author2=Jakeman, E. |author3=Matthews, J. O. |title=Generation and monitoring of a discrete stable random process|journal=Journal of Physics A|date=2002|volume=35|issue=49|pages=L745–752|doi=10.1088/0305-4470/35/49/101|bibcode=2002JPhA...35L.745H}} through their probability-generating function

: $G(s| \nu,a)=\sum_{n=0}^\infty P(N| \nu,a)(1-s)^N = \exp(-a s^\nu).$

In the above, $a>0$ is a scale parameter and $0<\nu\le1$ describes the power-law behaviour such that when $0<\nu<1$ ,

: $\lim_{N \to \infty}P(N|\nu,a) \sim \frac{1}{N^{\nu+1}}.$

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

The characteristic function of a discrete-stable distribution has the form{{cite web|title=Modeling financial returns by discrete stable distributions|author1=Slamova, Lenka|author2=Klebanov, Lev|url=http://mme2012.opf.slu.cz/proceedings/pdf/138_Slamova.pdf|publisher=International Conference Mathematical Methods in Economics|accessdate=2023-07-07}}

: $\varphi(t; a, \nu) = \exp \left[a \left( e^{it} - 1 \right)^\nu \right]$ , with $a>0$ and $0<\nu\le1$ .

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:

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

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

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

Expressions exist, however, that use special functions for the case $\nu=1/2$ {{cite journal|last=Matthews, J. O.|author2=Hopcraft, K. I. |author3=Jakeman, E. |title=Generation and monitoring of discrete stable random processes using multiple immigration population models|journal=Journal of Physics A|date=2003|volume=36|issue=46 |pages=11585–11603|doi=10.1088/0305-4470/36/46/004|bibcode=2003JPhA...3611585M}} (in terms of Bessel functions) and $\nu=1/3$ {{Cite thesis

|degree=PhD |title=Continuous and discrete properties of stochastic processes

|url=http://etheses.nottingham.ac.uk/1194/

|last=Lee |first=W.H. |year=2010

|publisher=The University of Nottingham

}} (in terms of hypergeometric functions).

As compound probability distributions

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

: $P(N| \alpha, c \sec( \alpha \pi / 2)) =
\int_0^\infty P(N| 1, \lambda)p(\lambda; \alpha, 1, c, 0) \, d\lambda$

where $p(x; \alpha, 1, c, 0)$ is the pdf of a one-sided continuous-stable distribution with symmetry parameter $\beta=1$ 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 $\nu \cdot \alpha$ and reduces the power-law index of the original distribution by a factor of $\alpha$ .

In other words,

: $P(N| \nu \cdot \alpha, c \sec(\pi \alpha / 2)) =
\int_0^\infty P(N| \alpha, \lambda)p(\lambda; \nu, 1, c, 0) \, d\lambda.$

Poisson limit

In the limit $\nu \rarr 1$ , the discrete-stable distributions behave{{cite journal|last=Lee, W. H.|author2=Hopcraft, K. I. |author3=Jakeman, E. |title=Continuous and discrete stable processes|journal=Physical Review E|date=2008|volume=77|issue=1|pages=011109–1 to 011109–04|doi=10.1103/PhysRevE.77.011109|pmid=18351820 |bibcode=2008PhRvE..77a1109L}} like a Poisson distribution with mean $a \sec(\nu \pi / 2)$ for small $N$ , but for $N \gg 1$ , the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails $P(N) \sim 1/N^{1 + \nu}$ to a discrete-stable distribution is extraordinarily slow{{cite journal|last=Hopcraft, K. I.|author2=Jakeman, E. |author3=Matthews, J. O. |title=Discrete scale-free distributions and associated limit theorems|journal=Journal of Physics A|date=2004|volume=37|issue=48|pages=L635–L642|doi=10.1088/0305-4470/37/48/L01|bibcode=2004JPhA...37L.635H}} when $\nu \approx 1$ , the limit being the Poisson distribution when $\nu > 1$ and $P(N| \nu, a)$ when $\nu \leq 1$ .

Discrete-stable distribution

Definition

As compound probability distributions

Poisson limit

See also

References

Further reading