On asymptotic properties of high moments of compound Poisson distribution

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.