Open Access
2016 On moment sequences and mixed Poisson distributions
Markus Kuba, Alois Panholzer
Probab. Surveys 13: 89-155 (2016). DOI: 10.1214/14-PS244


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 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.


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Markus Kuba. Alois Panholzer. "On moment sequences and mixed Poisson distributions." Probab. Surveys 13 89 - 155, 2016.


Received: 1 September 2014; Published: 2016
First available in Project Euclid: 20 September 2016

zbMATH: 1377.60021
MathSciNet: MR3550936
Digital Object Identifier: 10.1214/14-PS244

Primary: 60C05

Keywords: factorial moments , limiting distributions , Mixed Poisson distribution , parking functions , record-subtrees , Stirling transform , urn models

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.13 • 2016
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