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 Teor. Veroyatnost. i Primenen., 2016, Volume 61, Issue 2, Pages 234–267 (Mi tvp5055)

On the time of attaining a high level by a transient random walk in a random environment

V. I. Afanasyev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let a sequence of independent identically distributed pairs of random variables $(p_{i},q_{i})$, $i\in \mathbf{Z}$, be given, with ${p_{0}+q_{0}=1}$ and $p_{0}>0$, $q_{0}>0$ a.s. We consider a random walk in the random environment $(p_{i},q_{i})$, $i\in \mathbf{Z}$. This means that under a fixed environment a walking particle located at some moment in a state $i$ jumps either to the state $(i+1)$ with probability $p_{i}$ or to the state $(i-1)$ with probability $q_{i}$. It is assumed that $\mathbf{E} \log (p_{0}/q_{0}) <0$, i.e., that the random walk tends with time to $-\infty$. The set of such random walks may be divided into three types according to the value of the quantity $\mathbf{E} ((p_{0}/q_{0}) \log (p_{0}/q_{0}))$. In the case when the expectation above is zero we prove a limit theorem as $n\to \infty$ for the of time distribution of reaching the level $n$ by the mentioned random walk.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00318_à

DOI: https://doi.org/10.4213/tvp5055

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English version:
Theory of Probability and its Applications, 2017, 61:2, 178–207

Bibliographic databases:

Document Type: Article
Revised: 21.09.2015

Citation: V. I. Afanasyev, “On the time of attaining a high level by a transient random walk in a random environment”, Teor. Veroyatnost. i Primenen., 61:2 (2016), 234–267; Theory Probab. Appl., 61:2 (2017), 178–207

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp5055
• https://doi.org/10.4213/tvp5055
• http://mi.mathnet.ru/eng/tvp/v61/i2/p234

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This publication is cited in the following articles:
1. V. I. Afanasyev, “On the non-recurrent random walk in a random environment”, Discrete Math. Appl., 28:3 (2018), 139–156
2. V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Theory Probab. Appl., 63:3 (2019), 339–350
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