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 $\alpha \gt 1$ and $l$ is a positive function slowly varying at $\infty$. The same conclusion is also proved for the maximum of the martingale $W^{\ast} = \sup_{n \geq 1} W_n$ 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.

中文翻译：

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随机环境中超临界分支过程的加权加权矩

我们考虑在独立且均匀分布的随机环境$ \ xi =（\ xi_n）$中的超临界分支过程$（Z_n）$。令$ W $为自然mar的极限$ W_n = Z_n / E_ \ xi Z_n，n \ geq 0 $，其中$ E_ \ xi $表示给定环境$ \ xi $的条件期望。我们找到了存在形式为$ E_ \ xi W ^ {\ alpha} l（W）$的$ W $猝灭加权矩的必要和充分条件，其中$ \ alpha \ gt 1 $和$ l $为正函数在$ \ infty $处缓慢变化。对于the $ W ^ {\ ast} = \ sup_ {n \ geq 1} W_n $的最大值，而不是极限变量$ W $，也证明了相同的结论。在证明中，我们首先展示了Doob关于非负子集市加权矩的不等式的扩展形式，这是具有独立利益的。