On Sibuya-like distributions in branching and birth-and-death processes

TL;DR

This paper explores properties of heavy-tailed Sibuya-like distributions, extending thinning operations, and identifying their role in branching and birth-death processes, including new classes of distributions and moment conditions.

Contribution

It introduces a new class of distributions via scaled thinning, connects Sibuya-like distributions to branching processes, and characterizes moment finiteness for heavy-tailed variables.

Findings

Identification of a new distribution class with Laplace transform representation

Connection of Sibuya distributions to solutions of birth-death equations

Conditions for finite moments of heavy-tailed distributions

Abstract

We report some properties of heavy-tailed Sibuya-like distributions related to thinning, self-decomposability and branching processes. Extension of the thinning operation of on-negative integer-valued random variables to scaling by arbitrary positive number leads to a new class of probability distributions with generating function $Q(w)$ expressible as a Laplace transform $\varphi(1-w)$ and probability mass function $p_n$ satisfying simple one step recurrence relation between $p_{n+1}$ and $p_n$. We show that the compound Poisson-Sibuya and the shifted Sibuya distributions belong to this class. Using the fact that the same Markov property is present in stationary solutions of the birth and death equations we identify the Sibuya distribution and some of its variants as particular solutions of these equations. We also establish condition when integer-valued non-negative heavy-tailed random variable has finite $r$-th absolute moment ($0 < r < a < 1$).