Geodesic
Critical branching processes in random environment: the probability of extinction at a given moment
Diskretnaya Matematika, Tome 9 (1997) no. 4, pp. 100-126

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Let $\kappa$ be the extinction moment of a critical branching process in random environment with linear-fractional generating functions $f_n(s)$ of the number of particles in the $n$th generation, $n=1,2,\dots$ We give conditions on the parameters of $f_n(s)$ under which $$ \mathsf P\{\kappa=n\}\sim cn^{-3/2}, $$ as $n\to\infty$, where $c$ is a positive constant. This research was supported by the Russian Foundation for Basic Research, grant 96–01–00338, and by INTAS–RFBR, grant 95–0099. 
@article{DM_1997_9_4_a10,
     author = {V. A. Vatutin and E. E. D'yakonova},
     title = {Critical branching processes in random environment: the probability of extinction at a given moment},
     journal = {Diskretnaya Matematika},
     pages = {100--126},
     publisher = {mathdoc},
     volume = {9},
     number = {4},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1997_9_4_a10/}
}
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V. A. Vatutin; E. E. D'yakonova. Critical branching processes in random environment: the probability of extinction at a given moment. Diskretnaya Matematika, Tome 9 (1997) no. 4, pp. 100-126. http://geodesic.mathdoc.fr/item/DM_1997_9_4_a10/