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In this article we survey properties of mixed Poisson distributions andprobabilistic aspects of the Stirling transform: given a non-negative randomvariable $X$ with moment sequence $(\mu_s)_{s\in\mathbb{N}}$ we determine adiscrete random variable $Y$, whose moment sequence is given by the Stirlingtransform of the sequence $(\mu_s)_{s\in\mathbb{N}}$, and identify thedistribution as a mixed Poisson distribution. We discuss properties of thisfamily of distributions and present a simple limit theorem based on expansionsof factorial moments instead of power moments. Moreover, we present severalexamples of mixed Poisson distributions in the analysis of random discretestructures, unifying and extending earlier results. We also add severalentirely new results: we analyse triangular urn models, where the initialconfiguration or the dimension of the urn is not fixed, but may depend on thediscrete time $n$. We discuss the branching structure of plane recursive treesand its relation to table sizes in the Chinese restaurant process. Furthermore,we discuss root isolation procedures in Cayley-trees, a parameter in parkingfunctions, zero contacts in lattice paths consisting of bridges, and aparameter related to cyclic points and trees in graphs of random mappings, allleading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixedPoisson distributions naturally arise in the critical composition scheme ofAnalytic Combinatorics.

On moment sequences and mixed Poisson distributions