Geodesic
On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk
Diskretnaya Matematika, Tome 12 (2000) no. 2, pp. 31-50

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Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions, $T$ be the time of extinction of $\{\xi_n\}$, $T_M$ be the first maximum passage time of $\{\xi_n\}$. We study the asymptotic behaviour of $\mathsf P(T_M>n)$ and prove limit theorems for the random variables $\{T_M/n\mid T>n\}$ and $\{T_M/T\mid T>n\}$ as $n\to\infty$. Similar results are established for the stopped random walk with zero drift. 
@article{DM_2000_12_2_a2,
     author = {V. I. Afanasyev},
     title = {On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk},
     journal = {Diskretnaya Matematika},
     pages = {31--50},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2000_12_2_a2/}
}
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V. I. Afanasyev. On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk. Diskretnaya Matematika, Tome 12 (2000) no. 2, pp. 31-50. http://geodesic.mathdoc.fr/item/DM_2000_12_2_a2/