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Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 3, Pages 417–430 (Mi tvp5192)  

This article is cited in 1 scientific paper (total in 1 paper)

Two-boundary problem for a random walk in a random environment

V. I. Afanasyev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Given a sequence of independent identically distributed pairs of random variables $(p_i,q_i)$, $i\in\mathbf{Z}$, with $p_0+q_0=1$, and $p_0>0$ a.s., $q_0>0$ a.s., one considers a random walk in the random environment $(p_i,q_i)$, $i\in\mathbf{Z}$. This means that, for a fixed random environment, a walking particle transits from the state $i$ either to the state $(i+1)$ with probability $p_i$ or to the state $(i-1)$ with probability $q_i$. It is assumed that $\mathbf{E}\ln (p_0/q_0)=0$, that is, the walk is oscillating. We are concerned with the exit problem of the walk under consideration from the interval $(-\lfloor an\rfloor,\lfloor bn\rfloor)$, where $a$, $b$ are arbitrary positive constants. We find the asymptotics of the exit probability of the walk from the above interval from the right (the left). A limit theorem for the exit time of the walk from this interval is obtained.

Keywords: random walk in random environment, branching process 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/tvp5192

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English version:
Theory of Probability and its Applications, 2019, 63:3, 339–350

Bibliographic databases:

Received: 19.11.2017
Revised: 21.02.2018
Accepted:06.03.2018

Citation: V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Teor. Veroyatnost. i Primenen., 63:3 (2018), 417–430; Theory Probab. Appl., 63:3 (2019), 339–350

Citation in format AMSBIB
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  • Теория вероятностей и ее применения Theory of Probability and its Applications
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