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Teor. Veroyatnost. i Primenen., 2012, Volume 57, Issue 4, Pages 625–648 (Mi tvp4471)  

This article is cited in 6 scientific papers (total in 6 papers)

About time of reaching a high level by a random walk in a random environment

V. I. Afanasyev

Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: Let $(p_{i},q_{i}) $, $i\in \mathbb{Z}$, be a sequence of independent identically distributed pairs of random variables, where $p_{0}+q_{0}=1$ and, in addition, $p_{0}>0$ and $q_{0}>0 $ a.s. We consider a random walk in the random environment $(p_{i},q_{i}) $, $i\in \mathbb{Z}$. This means that in a fixed random environment a walking particle located at some moment $n$ at a state $i$ jumps at moment $n+1$ either to the state $(i+1)$ with probability $p_{i}$ or to the state $(i-1)$ with probability $q_{i}$. It is assumed that the distribution of the random variable $\log (q_{0}/p_{0})$ belongs (without centering) to the domain of attraction of the two-sided stable law with index $\alpha \in (0,2] $. Let $T_{n}$ be the first passage time of level $n$ by the mentioned random walk. We prove the invariance principle for the logarithm of the stochastic process $\{T_{\lfloor ns\rfloor},s\in [0,1] \}$ as $n\to \infty$. This result is based on the limit theorem for a branching process in a random environment which allows precisely one immigrant in each generation.

Keywords: random walk in random environment, branching process in random environment with immigration, functional limit theorems, stable Lévy processes.

Funding Agency Grant Number
Russian Foundation for Basic Research 08-01-91954
Russian Academy of Sciences - Federal Agency for Scientific Organizations


DOI: https://doi.org/10.4213/tvp4471

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English version:
Theory of Probability and its Applications, 2013, 57:4, 547–567

Bibliographic databases:

Document Type: Article
MSC: 60G50,60K37,60F17,60J80
Received: 01.06.2010
Revised: 30.08.2012

Citation: V. I. Afanasyev, “About time of reaching a high level by a random walk in a random environment”, Teor. Veroyatnost. i Primenen., 57:4 (2012), 625–648; Theory Probab. Appl., 57:4 (2013), 547–567

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Afanasyev, “Conditional limit theorem for maximum of random walk in a random environment”, Theory Probab. Appl., 58:4 (2014), 525–545  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. V. I. Afanasyev, “On the time of attaining a high level by a transient random walk in a random environment”, Theory Probab. Appl., 61:2 (2017), 178–207  mathnet  crossref  crossref  mathscinet  isi  elib
    3. 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
    4. Hong W. Wang H., “Branching Structures Within Random Walks and Their Applications”, Branching Processes and Their Applications, Lecture Notes in Statistics, 219, ed. DelPuerto I. Gonzalez M. Gutierrez C. Martinez R. Minuesa C. Molina M. Mota M. Ramos A., Springer, 2016, 57–73  crossref  mathscinet  zmath  isi
    5. 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
    6. “Tezisy dokladov, predstavlennykh na Tretei Mezhdunarodnoi konferentsii po stokhasticheskim metodam”, Teoriya veroyatn. i ee primen., 64:1 (2019), 151–204  mathnet  crossref  elib
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