Geodesic
On the time of reaching a fixed level by a critical branching process in a random environment
Diskretnaya Matematika, Tome 11 (1999) no. 4, pp. 33-47

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Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions; let $T$ be the extinction time of $\{\xi_n\}$, and $T_x$ be the time of first passage of the semiaxis $(x,\infty)$. We find the asymptotic distributions of the random variables $T_x/\ln^2 x$, $T_x/T$, $T/\ln^2x$ under the condition $\{T_x\infty\}$ as $x\to \infty$. 
@article{DM_1999_11_4_a2,
     author = {V. I. Afanasyev},
     title = {On the time of reaching a fixed level by a critical branching process in a random environment},
     journal = {Diskretnaya Matematika},
     pages = {33--47},
     publisher = {mathdoc},
     volume = {11},
     number = {4},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1999_11_4_a2/}
}
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V. I. Afanasyev. On the time of reaching a fixed level by a critical branching process in a random environment. Diskretnaya Matematika, Tome 11 (1999) no. 4, pp. 33-47. http://geodesic.mathdoc.fr/item/DM_1999_11_4_a2/