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 mean .
The characteristic function of a discrete-stable distribution has the form:[6]
- , with and .
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 do exist, however, using 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 distributions 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 stability parameter and scale parameter the resultant distribution is[9] discrete-stable with index and scale parameter .
Formally, this is written:
- ;\alpha ,1,c,0)\,d\lambda }
where is the pdf of a one-sided continuous-stable distribution with symmetry paramètre 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 , reducing the power-law index of the original distribution by a factor of .
In other words,
- ;\nu ,1,c,0)\,d\lambda .}
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.