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

This article is cited in 2 scientific papers (total in 2 papers)

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_а


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
Received: 14.01.2015
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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  crossref  mathscinet  isi  elib
    2. V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Theory Probab. Appl., 63:3 (2019), 339–350  mathnet  crossref  crossref  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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