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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 continuous-stable distributions.
Discrete-stable distributions have been used in numerous fields, in particular in scale-free networks such as the internet and social networks [2] or even semantic networks. [3]
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. [4] It is the only discrete-stable distribution for which the mean and all higher-order moments are finite.[ dubious – discuss ]
The discrete-stable distributions are defined [5] through their probability-generating function
In the above, is a scale parameter and describes the power-law behaviour such that when ,
When , the distribution becomes the familiar Poisson distribution with the mean .
The characteristic function of a discrete-stable distribution has the form [6]
Again, when , the distribution becomes the Poisson distribution with mean .
The original distribution is recovered through repeated differentiation of the generating function:
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
Expressions exist, however, that use special functions for the case [7] (in terms of Bessel functions) and [8] (in terms of hypergeometric functions).
The entire class of discrete-stable distributions can be formed as Poisson compound probability distribution where the mean, , 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 and scale parameter , the resultant distribution is [9] discrete-stable with index and scale parameter .
Formally, this is written
where is the pdf of a one-sided continuous-stable distribution with symmetry parameter and location parameter .
A more general result [8] states that forming a compound distribution from any discrete-stable distribution with index with a one-sided continuous-stable distribution with index results in a discrete-stable distribution with index and reduces the power-law index of the original distribution by a factor of .
In other words,
In the limit , the discrete-stable distributions behave [9] like a Poisson distribution with mean for small , but for , the power-law tail dominates.
The convergence of i.i.d. random variates with power-law tails to a discrete-stable distribution is extraordinarily slow [10] when , the limit being the Poisson distribution when and when .