RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 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., 2012, Volume 57, Issue 4, Pages 625–648 (Mi tvp4471)

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

Full text: PDF file (244 kB)
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
MSC: 60G50,60K37,60F17,60J80
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

Citation in format AMSBIB
\Bibitem{Afa12}
\by V.~I.~Afanasyev
\paper About time of reaching a high level by a random walk in a random environment
\jour Teor. Veroyatnost. i Primenen.
\yr 2012
\vol 57
\issue 4
\pages 625--648
\mathnet{http://mi.mathnet.ru/tvp4471}
\crossref{https://doi.org/10.4213/tvp4471}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3201664}
\zmath{https://zbmath.org/?q=an:1284.60087}
\elib{http://elibrary.ru/item.asp?id=20732979}
\transl
\jour Theory Probab. Appl.
\yr 2013
\vol 57
\issue 4
\pages 547--567
\crossref{https://doi.org/10.1137/S0040585X97986175}
\elib{http://elibrary.ru/item.asp?id=21887783}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84887188912}

• http://mi.mathnet.ru/eng/tvp4471
• https://doi.org/10.4213/tvp4471
• http://mi.mathnet.ru/eng/tvp/v57/i4/p625

 SHARE:

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, “Conditional limit theorem for maximum of random walk in a random environment”, Theory Probab. Appl., 58:4 (2014), 525–545
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
3. V. I. Afanasyev, “On the non-recurrent random walk in a random environment”, Discrete Math. Appl., 28:3 (2018), 139–156
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
5. V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Theory Probab. Appl., 63:3 (2019), 339–350
6. “Tezisy dokladov, predstavlennykh na Tretei Mezhdunarodnoi konferentsii po stokhasticheskim metodam”, Teoriya veroyatn. i ee primen., 64:1 (2019), 151–204
•  Number of views: This page: 287 Full text: 32 References: 42 First page: 4