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 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^[2] or even semantic networks.^[3]

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.^[4] 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^[5] through their probability-generating function

${\displaystyle G(s|\nu ,a)=\sum _{n=0}^{\infty }P(N|\nu ,a)(1-s)^{N}=\exp(-as^{\nu }).}$

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

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

When ${\displaystyle \nu =1}$ the distribution becomes the familiar Poisson distribution with mean ${\displaystyle a}$.

The characteristic function of a discrete-stable distribution has the form:^[6]

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

Again, when ${\displaystyle \nu =1}$ the distribution becomes the Poisson distribution with mean ${\displaystyle a}$.

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

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

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

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

Expressions do exist, however, using special functions for the case ${\displaystyle \nu =1/2}$^[7] (in terms of Bessel functions) and ${\displaystyle \nu =1/3}$^[8] (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, ${\displaystyle \lambda }$, 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 ${\displaystyle 0<\alpha <1}$ and scale parameter ${\displaystyle c}$ the resultant distribution is^[9] discrete-stable with index ${\displaystyle \nu =\alpha }$ and scale parameter ${\displaystyle a=c\sec(\alpha \pi /2)}$.

Formally, this is written:

${\displaystyle P(N|\alpha ,c\sec(\alpha \pi /2))=\int _{0}^{\infty }P(N|1,\lambda )p(\lambda$ ;\alpha ,1,c,0)\,d\lambda }

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

A more general result^[8] states that forming a compound distribution from any discrete-stable distribution with index ${\displaystyle \nu }$ with a one-sided continuous-stable distribution with index ${\displaystyle \alpha }$ results in a discrete-stable distribution with index ${\displaystyle \nu \cdot \alpha }$, reducing the power-law index of the original distribution by a factor of ${\displaystyle \alpha }$.

In other words,

${\displaystyle P(N|\nu \cdot \alpha ,c\sec(\pi \alpha /2))=\int _{0}^{\infty }P(N|\alpha ,\lambda )p(\lambda$ ;\nu ,1,c,0)\,d\lambda .}

In the Poisson limit

In the limit ${\displaystyle \nu \rightarrow 1}$, the discrete-stable distributions behave^[9] like a Poisson distribution with mean ${\displaystyle a\sec(\nu \pi /2)}$ for small ${\displaystyle N}$, however for ${\displaystyle N\gg 1}$, the power-law tail dominates.

The convergence of i.i.d. random variates with power-law tails ${\displaystyle P(N)\sim 1/N^{1+\nu }}$ to a discrete-stable distribution is extraordinarily slow^[10] when ${\displaystyle \nu \approx 1}$ - the limit being the Poisson distribution when ${\displaystyle \nu >1}$ and ${\displaystyle P(N|\nu ,a)}$ when ${\displaystyle \nu \leq 1}$.

See also

• Stable distribution
• Poisson distribution

References

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.
• Ibragimov, I.; Linnik, Yu (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands.