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)_{sinmathbb{N}}) we determine a discrete random variable (Y), whose moment sequence is given by the Stirling transform of the sequence ((mu_s)_{sinmathbb{N}}), and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a new 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. <script type=text/javascript src=//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML>
Highlights
In combinatorics the Stirling transform of a given sequences∈N, see [12, 75], is the sequences∈N, with elements given by ss bs = k ak, s ≥ 1. (1) k=1The inverse Stirling transform of the sequences∈N is obtained as follows: as =s (−1)s−k s k bk, s ≥ 1. (2)Here s k denote the Stirling numbers of the second kind, counting the number of ways to partition a set of s objects into k non-empty subsets, see [73]or [30], and s k denotes the unsigned Stirling numbers of the first kind, Date: November 15, 2021. 2000 Mathematics Subject Classification. 60C05
A random variable Y with factorial moments given by E(Y s) = ρsμs has a mixed Poisson distribution Y =L MPo(ρX) with mixing distribution X and scale parameter ρ > 0, and the sequence of power moments of Y is the Stirling transform of the moment sequences∈N
Let (Xn)n∈N denote a sequence of random variables, whose factorial moments are asymptotically of mixed Poisson type satisfying for n tending to infinity the asymptotic expansion with μs ≥ 0, and λn > 0
Summary
Mixed Poisson distribution, Factorial moments, Stirling transform, Limiting distributions, Urn models, Parking functions, Record-subtrees. The aim of this work is to discuss several probabilistic aspects of a generalized Stirling transform with parameter ρ > 0 in connection with moment sequences and mixed Poisson distributions, pointing out applications in the analysis of random discrete structures. Our main motivation to study random variables with a given sequence of factorial moments (6) stems from the analysis of combinatorial structures. The goal of this work is twofold: first, to survey the properties of mixed Poisson distributions, and second to discuss their appearances in combinatorics and the analysis of random discrete structures, complementing existing results; we will add several entirely new results.
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