Discrete-stable distribution

The discrete-stable distributions 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 or even semantic networks.

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. 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 through their probability-generating function

[math]\displaystyle{ G(s| \nu,a)=\sum_{n=0}^\infty P(N| \nu,a)(1-s)^N = \exp(-a s^\nu). }[/math]

In the above, [math]\displaystyle{ a\gt 0 }[/math] is a scale parameter and [math]\displaystyle{ 0\lt \nu\le1 }[/math] describes the power-law behaviour such that when [math]\displaystyle{ 0\lt \nu\lt 1 }[/math],

[math]\displaystyle{ \lim_{N \to \infty}P(N|\nu,a) \sim \frac{1}{N^{\nu+1}}. }[/math]

When [math]\displaystyle{ \nu=1 }[/math] the distribution becomes the familiar Poisson distribution with mean [math]\displaystyle{ a }[/math].

The characteristic function of a discrete-stable distribution has the form:

[math]\displaystyle{ \varphi(t; a, \nu) = \exp \left[a \left( e^{it} - 1 \right)^\nu \right] }[/math], with [math]\displaystyle{ a\gt 0 }[/math] and [math]\displaystyle{ 0\lt \nu\le1 }[/math].

Again, when [math]\displaystyle{ \nu=1 }[/math] the distribution becomes the Poisson distribution with mean [math]\displaystyle{ a }[/math].

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

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

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

[math]\displaystyle{ \!P(N| \nu=1, a)= \frac{a^N e^{-a}}{N!}. }[/math]

Expressions do exist, however, using special functions for the case [math]\displaystyle{ \nu=1/2 }[/math] (in terms of Bessel functions) and [math]\displaystyle{ \nu=1/3 }[/math] (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, [math]\displaystyle{ \lambda }[/math], 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 [math]\displaystyle{ 0 \lt \alpha \lt 1 }[/math] and scale parameter [math]\displaystyle{ c }[/math] the resultant distribution is discrete-stable with index [math]\displaystyle{ \nu = \alpha }[/math] and scale parameter [math]\displaystyle{ a = c \sec( \alpha \pi / 2) }[/math].

Formally, this is written:

[math]\displaystyle{ P(N| \alpha, c \sec( \alpha \pi / 2)) = \int_0^\infty P(N| 1, \lambda)p(\lambda; \alpha, 1, c, 0) \, d\lambda }[/math]

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

A more general result states that forming a compound distribution from any discrete-stable distribution with index [math]\displaystyle{ \nu }[/math] with a one-sided continuous-stable distribution with index [math]\displaystyle{ \alpha }[/math] results in a discrete-stable distribution with index [math]\displaystyle{ \nu \cdot \alpha }[/math], reducing the power-law index of the original distribution by a factor of [math]\displaystyle{ \alpha }[/math].

In other words,

[math]\displaystyle{ 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. }[/math]

In the Poisson limit

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

The convergence of i.i.d. random variates with power-law tails [math]\displaystyle{ P(N) \sim 1/N^{1 + \nu} }[/math] to a discrete-stable distribution is extraordinarily slow when [math]\displaystyle{ \nu \approx 1 }[/math] - the limit being the Poisson distribution when [math]\displaystyle{ \nu \gt 1 }[/math] and [math]\displaystyle{ P(N| \nu, a) }[/math] when [math]\displaystyle{ \nu \leq 1 }[/math].

See also

• Stable distribution
• Poisson distribution

Further reading

• Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley. ISBN:0-471-25709-5
• Gnedenko, B. V.; Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley. https://archive.org/details/limitdistributio00gned_0. 
• Ibragimov, I.; Linnik, Yu (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands.