Quenched weighted moments of a supercritical branching process in a random environment

We consider a supercritical branching process $(Z_n)$ in an independent and identically distributed random environment $\xi =(\xi_n)$. Let $W$ be the limit of the natural martingale $W_n = Z_n / E_\xi Z_n (n \geq 0)$, where $E_\xi $ denotes the conditional expectation given the environment $\xi$. We find a necessary and sufficient condition for the existence of quenched weighted moments of $W$ of the form $E_{\xi} W^{\alpha} l(W)$, where $\alpha > 1$ and $l$ is a positive function slowly varying at $\infty$. The same conclusion is also proved for the maximum of the martingale $W^* = \sup_{n\geq 1} W_n $ instead of the limit variable $W$. In the proof we first show an extended version of Doob's inequality about weighted moments for nonnegative submartingales, which is of independent interest.