Teoriya Veroyatnostei i ee Primeneniya
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 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 This work was supported by the program “Dynamical systems and control theory” of the Presidium of RAS (grant PRAS-18-01).

DOI: https://doi.org/10.4213/tvp5192

Full text: PDF file (480 kB)
References: PDF file   HTML file

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} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3833090} \elib{https://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{https://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

 SHARE:

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

Related presentations:

This publication is cited in the following articles:
1. Smadi Ch., Vatutin V., “Critical Branching Processes in Random Environment With Immigration: Survival of a Single Family”, Extremes
2. “Abstracts of talks given at the 3rd International Conference on Stochastic Methods”, Theory Probab. Appl., 64:1 (2019), 124–169
3. V. I. Afanasyev, “On the times of attaining high levels by a random walk in a random environment”, Theory Probab. Appl., 65:3 (2020), 359–374
4. V. A. Vatutin, E. E. D'yakonova, “Subcritical branching processes in random environment with immigration: Survival of a single family”, Theory Probab. Appl., 65:4 (2021), 527–544
•  Number of views: This page: 351 Full text: 18 References: 29 First page: 17