Weighted moments of the limit of a branching process in a random environment

Abstract

Let $(Z_n)$ be a supercritical branching process in a random environment , and $W$ be the limit of the normalized population size $Z_n/\mathbb{E%}(Z_n|ζ)$. We show necessary and sufficient conditions for the existence of weighted moments of $W$ of the form $\E W^α\ell(W)$, where $\alpha \ge 1$, $\ell$ is a positive function slowly varying at $\infty$. In the Galton-Watson case, the results improve those of Bingham and Doney (1974).

arXiv comment16 pages