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

Xingang Liang ; Quansheng Liu

Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 135-153.

Résumé

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\zeta =({\zeta }_0,{\zeta }_1,\dots )$, 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 $\alpha \ge 1$ and $\ell$ is a positive function slowly varying at $\mathrm{\infty }$.

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