On moment sequences and mixed Poisson distributions

TL;DR

This paper surveys properties of mixed Poisson distributions, explores their connections with Stirling transforms, and presents new results on their applications in combinatorics, random structures, and urn models, including limit theorems and distributional analyses.

Contribution

It introduces new analytical results on mixed Poisson distributions, including limit theorems, and applies them to various combinatorial and probabilistic models, extending previous work.

Findings

Identified mixed Poisson distributions via Stirling transforms.

Derived a simple limit theorem based on factorial moments.

Analyzed applications in urn models, trees, parking functions, and random mappings.

Abstract

In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable $X$ with moment sequence $(\mu_s)_{s\in\mathbb{N}}$ we determine a discrete random variable $Y$, whose moment sequence is given by the Stirling transform of the sequence $(\mu_s)_{s\in\mathbb{N}}$, and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time $n$. We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley-trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics.