  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 $\lambda= \lambda_k$.
