Geodesic
On the times of attaining high levels by a random walk in a random environment
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 3, pp. 460-478 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $(p_i,q_i)$, $i\in \mathbf{Z}$, be a sequence of independent identically distributed random vectors such that $p_i,q_i>0$ and $p_i+q_i$ $=1$ a.s. for $i\in \mathbf{Z}$. We consider a random walk in the random environment $\{(p_i,q_i)$, $i\in \mathbf{Z}\}$. It is assumed that $\mathbf{E}\ln (p_0/q_0)=0$ and $0<\mathbf{E}\ln^{2}(q_0/p_0)<+\infty$. We study the times of attaining $T_{n_1},\dots,T_{n_m}$ of increasing levels $n_1,\dots,n_m$ of order $n$. It is proved that the underlying probability space can be partitioned into random events (depending on $n$) such that their probabilities for large $n$ are close to positive numbers, and on each such event, the set of times $T_{n_1},\dots,T_{n_m}$ is partitioned into consecutive groups such that elements of each group have the same order and are negligible compared with those of the successive group.
Keywords: random walk in random environment, branching in random environment with immigration, limit theorem. 
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V. I. Afanasyev. On the times of attaining high levels by a random walk in a random environment. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 3, pp. 460-478. http://geodesic.mathdoc.fr/item/TVP_2020_65_3_a1/

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