Discrete semi-stable distributions
The author introduces and studies a concept of semi-stability for discrete distributions on the non-negative integers as a generalization of discrete stable distributions, which parallels the concept of semi-stable distributions on the real line. For discrete semi-stable random variables, a functional equation for the probability generating function is required to be fulfilled for a single parameter only. Discrete semi-stable distributions are shown to be infinitely divisible and to admit a certain canonical representation of their probability generating function. Further, a distributional equality using the Steutel and van Harn operator is shown to be characteristic for discrete semi-stability, from which the author deduces and studies discrete geometric semi-stable distributions by geometric randomization of a certain parameter, as well as corresponding limit theorems with discrete semi-stable limits. Examples based on Poisson mixtures with respect to a semi-stable distribution on the positive real line are also presented.
scientific article; zbMATH DE number 2144521
scientific article