On asymptotic properties of high moments of compound Poisson distribution

TL;DR

This paper investigates the asymptotic behavior of high moments of compound Poisson distributions, providing explicit formulas and applications to random graphs, random matrices, and combinatorial polynomials.

Contribution

It derives an explicit leading term for the moments as the order grows and analyzes its dependence on the Poisson parameter, with applications to various probabilistic and combinatorial models.

Findings

Explicit expression for the leading term of moments as k→∞

Concentration results for maximum vertex degree in large weighted graphs

Asymptotic analysis of Bell polynomials and related combinatorial polynomials

Abstract

We study asymptotic behavior of the moments $M_k(\lambda)$ of the sum $X_1+\dots+X_{N_\lambda}$, where $N_\lambda$ follows the Poisson probability distribution with mean value $\lambda$ and $\{X_j\}$ is a family of i.i.d. random variables also independent from $N_\lambda$. We obtain an explicit expression for the leading term of $M_k(\lambda)$ as $k\to\infty$ and study it in dependence of the asymptotic behavior of $\lambda= \lambda_k$. In application, we establish a concentration property of maximal vertex degree of large weighted random graphs. Another application is related with a variable that arises in the studies of high moments of large random matrices. Finally, regarding three particular cases of probability distribution of $X_j$, we comment on the asymptotic behavior of certain combinatorial polynomials, including the Bell polynomials of even partitions.