On moment sequences and mixed Poisson distributions

Abstract

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 (µs) s∈N we determine a discrete random variable Y , whose moment sequence is given by the Stirling transform of the sequence (µs) s∈N , and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a 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.

On bucket increasing trees, clustered increasing trees and increasing diamonds

“…We remark in passing that, alternatively, Y n, ,j can also be described via a generalized P ólya urn model; see [13].…”

On multisets, interpolated multiple zeta values and limit laws

“…Thus, the factorial moments are almost of mixed Poisson type [25] with standard exponential mixing distribution; the additional factor k n+k can be explained by the definition of the parameter σ, which influences the discrete limit case. This directly leads to the stated limit laws using Lemma 2 of [25]. Alternatively, the discrete limit for k/n → c can be directly obtained as follows:…”

Explicit characterization of moments of balanced triangular Pólya urns by an elementary approach

“…Such an exact moment calculation involves sums, with a number of summands that steadily increases as the ball drawing continues. The exact factorial moments have very recently computed in Kuba and Panholzer (2014), and it is not hard to extract the plain moments from them. However, the first two of these three references use rather sophisticated methods.…”

“…For instance, Janson (2006) uses branching process and martingales, while Flajolet et al (2006) uses analytic combinatorics based on systems of differential equations. On the other hand, Kuba and Panholzer (2014) uses a simpler method based on mixed Poisson distributions, and some elements in moment calculations are done in a similar spirit to the present study.…”

“…Moments of ball counts of a particular color have been obtained exactly and asymptotically in a number of sources (Janson, 2006, Kuba and Panholzer, 2014, and Flajolet et al, 2006. Exact factorial moments are in Kuba and Panholzer (2014). While Flajolet et al (2006) gives an exact distribution, finding the exact moments from it is daunting.…”