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Teor. Veroyatnost. i Primenen., 2020, Volume 65, Issue 3, Pages 460–478 (Mi tvp5307)  

On the times of attaining high levels by a random walk in a random environment

V. I. Afanasyev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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 $T_{n_1},…,T_{n_m}$ of increasing levels $n_1,…,n_m$ 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 $T_{n_1},…,T_{n_m}$ 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.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01


DOI: https://doi.org/10.4213/tvp5307

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English version:
Theory of Probability and its Applications, 2020, 65:3, 359–374

Bibliographic databases:

Received: 13.03.2019
Revised: 12.11.2019
Accepted:20.11.2019

Citation: V. I. Afanasyev, “On the times of attaining high levels by a random walk in a random environment”, Teor. Veroyatnost. i Primenen., 65:3 (2020), 460–478; Theory Probab. Appl., 65:3 (2020), 359–374

Citation in format AMSBIB
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