Discrete-stable distribution

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 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 classes of distribution have properties such as infinitely divisibility, power law tails and unimodality.

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

Definition

The discrete-stable distributions are defined^[4] 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 $0<\nu \leq 1$ 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 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 do exist, however, using special functions for the case $\nu =1/2$ ^[5] (in terms of Bessel functions) and $\nu=1/3$ ^[6] (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, $\lambda$ , 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 stability parameter $0<\alpha <1$ and scale parameter $c$ the resultant distribution is^[7] 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 paramètre $\beta =1$ and location parameter $\mu =0$ .

A more general result^[6] 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 $\nu \cdot \alpha$ , reducing 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.

In the Poisson limit

In the limit $\nu \rightarrow 1$ , the discrete-stable distributions behave^[7] 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 $P(N)\sim 1/N^{{1+\nu }}$ to a discrete-stable distribution is extraordinarily slow^[8] when $\nu \approx 1$ - the limit being the Poisson distribution when $\nu > 1$ and $P(N| \nu, a)$ when $\nu \leq 1$ .

References

↑ Steutel, F. W.; van Harn, K. (1979). "Discrete Analogues of Self-Decomposability and Stability". Annals of Probability. 7 (5): 893–899. doi:10.1214/aop/1176994950.
↑ 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.
↑ Steyvers, M.; Tenenbaum, J. B. (2005). "The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth". Cognitive science. 29 (1): 41–78. doi:10.1207/s15516709cog2901_3.
↑ Hopcraft, K. I.; Jakeman, E.; Matthews, J. O. (2002). "Generation and monitoring of a discrete stable random process". Journal of Physics A: General Physics. 35 (49): L745–752. doi:10.1088/0305-4470/35/49/101.
↑ Matthews, J. O.; Hopcraft, K. I.; Jakeman, E. (2003). "Generation and monitoring of discrete stable random processes using multiple immigration population models". Journal of Physics A: Mathematical and General. 36: 11585–11603. doi:10.1088/0305-4470/36/46/004.
1 2 Lee, W.H. (2010). Continuous and discrete properties of stochastic processes (PhD thesis). The University of Nottingham.
1 2 Lee, W. H.; Hopcraft, K. I.; Jakeman, E. (2008). "Continuous and discrete stable processes". Physical Review E. 77 (1): 011109–1 to 011109–04. doi:10.1103/PhysRevE.77.011109.
↑ Hopcraft, K. I.; Jakeman, E.; Matthews, J. O. (2004). "Discrete scale-free distributions and associated limit theorems". Journal of Physics A: General Physics. 37 (48): L635–L642. doi:10.1088/0305-4470/37/48/L01.

Discrete-stable distribution

Definition

As compound probability distributions

In the Poisson limit

See also

References

Further reading