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

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\zeta=(\zeta_0,\zeta_1,\ldots)$, and let $W$ be the limit of the normalized population size $Z_n/\mathbb E(Z_n|\zeta)$. We show a necessary and sufficient condition for the existence of weighted moments of $W$ of the form $\mathbb E\,W^\alpha\ell(W)$, where $\alpha\geq1$ and $\ell$ is a positive function slowly varying at $\infty$.