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M. Kuba, A. Panholzer:
"On Moment Sequences and Mixed Poisson Distributions";
Probability Surveys, 13 (2016), 89 - 155.

Abstract: In this article we survey properties of mixed Poisson distributions
and probabilistic aspects of the Stirling transform: given a nonnegative
random variable X with moment sequence (μ
s
)
we determine
a discrete random variable Y , whose moment sequence is given by the Stirling
transform of the sequence (μ
s
)
s∈N
s∈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.
MSC 2010 subject classifications: 60C05.
Keywords and phrases: Mixed Poisson distribution, factorial moments,

http://dx.doi.org/10.1214/14-PS244