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On moment sequences and mixed Poisson distributions. (English) Zbl 1377.60021
This survey article covers the basic theory as well as several applications of the mixed Poisson distribution. Roughly speaking, the mixed Poisson distribution refers to a class of distributions that follow the Poisson law whose parameter is a random variable. The distribution of this is called the mixing distribution. According to the choice of the mixing distribution, one can derive the ordinary Poisson distribution, the negative binomial distribution, the Rayleigh distribution and others. The article gives the basic theorems that associate the moments of the mixed Poisson distribution with the moments of the mixing distribution. These are obtained through the Stirling transformations that relate Stirling numbers of the first kind with those of the second kind. It also covers conditions that imply that a given sequence of random variables converges in distribution to the a random variable that follows the mixed Poisson distribution. After this fundamental theory, several applications of the mixed Poisson distribution are discussed. Those start with the block sizes of Stirling permutations, several urn models, such as the class of diminishing Pólya-Eggenberger urns models. Another important class of applications discussed is that of trees. In particular, increasing trees are discussed as well as recursive trees. In that context, the distribution of node degrees is considered. Thereafter, the model of triangular urns is considered, which is a generalisation of an urn model that is encountered in the analysis of random recursive trees. There, convergence in distribution is proved for the (rescaled) number of balls of a given colour. Also, a special class of mixed distribution with mixing distribution is discussed, namely the Rayleigh distribution. This occurs naturally in the context of random trees and parking functions. Particular applications are discussed. Also, some applications on the zeros of walks are discussed, as well as on random mappings. Finally, the notion of multivariate mixed Poisson distribution is discussed.

##### MSC:
 60C05 Combinatorial probability 60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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