Discrete Semi-Self-Decomposability Induced by Semigroups

Abstract

A continuous semigroup of probability generating functions $\mathcal{F}:=(F_t, t\ge 0)$ is used to introduce a notion of discrete semi-selfdecomposability, or $\mathcal{F}$-semi-selfdecomposability, for distributions with support on $\bf Z_+$. $\mathcal{F}$-semi-selfdecomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of $\mathcal{F}$-semi-selfdecomposable laws is shown to contain the $\mathcal{F}$- semistable distributions and the geometric $\mathcal{F}$-semistable distributions. A generalization of discrete random stability is also explored.