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Discrete stable distribution

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The discrete-stable distributions 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.

Contents

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 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 through their probability-generating function

G ( s | ν , a ) = ∑ n = 0 ∞ P ( N | ν , a ) ( 1 − s ) N = exp ⁡ ( − a s ν ) .

In the above, a > 0 is a scale parameter and 0 < ν ≤ 1 describes the power-law behaviour such that when 0 < ν < 1 ,

lim N → ∞ P ( N | ν , a ) ∼ 1 N ν + 1 .

When ν = 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 | ν , a ) = ( − 1 ) N N ! d N G ( s | ν , a ) d s N | 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 | ν = 1 , a ) = a N e − a N ! .

Expressions do exist, however, using special functions for the case ν = 1 / 2 (in terms of Bessel functions) and ν = 1 / 3 (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 < α < 1 and scale parameter c the resultant distribution is discrete-stable with index ν = α and scale parameter a = c sec ⁡ ( α π / 2 ) .

Formally, this is written:

P ( N | α , c sec ⁡ ( α π / 2 ) ) = ∫ 0 ∞ P ( N | 1 , λ ) p ( λ ; α , 1 , c , 0 ) d λ

where p ( x ; α , 1 , c , 0 ) is the pdf of a one-sided continuous-stable distribution with symmetry paramètre β = 1 and location parameter μ = 0 .

A more general result states that forming a compound distribution from any discrete-stable distribution with index ν with a one-sided continuous-stable distribution with index α results in a discrete-stable distribution with index ν ⋅ α , reducing the power-law index of the original distribution by a factor of α .

In other words,

P ( N | ν ⋅ α , c sec ⁡ ( π α / 2 ) = ∫ 0 ∞ P ( N | α , λ ) p ( λ ; ν , 1 , c , 0 ) d λ .

In the Poisson limit

In the limit ν → 1 , the discrete-stable distributions behave like a Poisson distribution with mean a sec ⁡ ( ν π / 2 ) for small N , however for N ≫ 1 , the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails P ( N ) ∼ 1 / N 1 + ν to a discrete-stable distribution is extraordinarily slow when ν ≈ 1 - the limit being the Poisson distribution when ν > 1 and P ( N | ν , a ) when ν ≤ 1 .

References

Discrete-stable distribution Wikipedia
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