Geodesic
Weighted moments of the limit of a~branching process in a~random environment
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 135-153

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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 $\infty$. 
@article{TM_2013_282_a11,
     author = {Xingang Liang and Quansheng Liu},
     title = {Weighted moments of the limit of a~branching process in a~random environment},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {135--153},
     publisher = {mathdoc},
     volume = {282},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_282_a11/}
}
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Xingang Liang; Quansheng Liu. Weighted moments of the limit of a~branching process in a~random environment. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 135-153. http://geodesic.mathdoc.fr/item/TM_2013_282_a11/