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

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