Broadly discrete stable distributions

Music City Center 

The central limit theorem shows that a suitably rescaled sum of random variables with finite variance converges to a Gaussian distribution. When the variance of the summands is infinite, the limiting distribution is instead a heavy tailed stable distribution, indexed by $\alpha\in (0,2]$ where $\alpha=2$ is the Gaussian special case. All nondegenerate stable distributions are absolutely continuous, but a discrete notion of stability can be obtained by replacing scaling with binomial thinning. Under this definition, it has been shown that the strictly discrete stable distributions are Poisson mixtures with maximally skewed stable mixing distributions for $\alpha\leq 1$. In previous work we have established the validity of Poisson-stable mixtures for $\alpha\in [1,2]$ despite the mixing distribution having support on negative numbers. Here we show that this mixed Poisson family is the unique set of distributions satisfying a broader definition of discrete stability with Poisson translation playing the role of subtraction. We also provide a compound Poisson representation and discuss discrete infinite divisibility.

probability

stable distributions

discrete stability

mixed Poisson

count data 

IMS