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

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:

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
\Bibitem{Afa18} \by V.~I.~Afanasyev \paper Two-boundary problem for a random walk in a random environment \jour Teor. Veroyatnost. i Primenen. \yr 2018 \vol 63 \issue 3 \pages 417--430 \mathnet{http://mi.mathnet.ru/tvp5192} \crossref{https://doi.org/10.4213/tvp5192} \elib{http://elibrary.ru/item.asp?id=35276549} \transl \jour Theory Probab. Appl. \yr 2019 \vol 63 \issue 3 \pages 339--350 \crossref{https://doi.org/10.1137/S0040585X97T98909X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457753200001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85064684530} 

• http://mi.mathnet.ru/eng/tvp5192
• https://doi.org/10.4213/tvp5192
• http://mi.mathnet.ru/eng/tvp/v63/i3/p417

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