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., 2020, Volume 65, Issue 3, Pages 460–478 (Mi tvp5307)

On the times of attaining high levels by a random walk in a random environment

V. I. Afanasyev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let $(p_i,q_i)$, $i\in \mathbf{Z}$, be a sequence of independent identically distributed random vectors such that $p_i,q_i>0$ and $p_i+q_i$ $=1$ a.s. for $i\in \mathbf{Z}$. We consider a random walk in the random environment $\{(p_i,q_i)$, $i\in \mathbf{Z}\}$. It is assumed that $\mathbf{E}\ln (p_0/q_0)=0$ and $0<\mathbf{E}\ln^{2}(q_0/p_0)<+\infty$. We study the times of attaining $T_{n_1},…,T_{n_m}$ of increasing levels $n_1,…,n_m$ of order $n$. It is proved that the underlying probability space can be partitioned into random events (depending on $n$) such that their probabilities for large $n$ are close to positive numbers, and on each such event, the set of times $T_{n_1},…,T_{n_m}$ is partitioned into consecutive groups such that elements of each group have the same order and are negligible compared with those of the successive group.

Keywords: random walk in random environment, branching 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/tvp5307

Full text: PDF file (442 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Theory of Probability and its Applications, 2020, 65:3, 359–374

Bibliographic databases:

Revised: 12.11.2019
Accepted:20.11.2019

Citation: V. I. Afanasyev, “On the times of attaining high levels by a random walk in a random environment”, Teor. Veroyatnost. i Primenen., 65:3 (2020), 460–478; Theory Probab. Appl., 65:3 (2020), 359–374

Citation in format AMSBIB
\Bibitem{Afa20} \by V.~I.~Afanasyev \paper On the times of attaining high levels by a random walk in a random environment \jour Teor. Veroyatnost. i Primenen. \yr 2020 \vol 65 \issue 3 \pages 460--478 \mathnet{http://mi.mathnet.ru/tvp5307} \crossref{https://doi.org/10.4213/tvp5307} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3699318} \elib{https://elibrary.ru/item.asp?id=45186591} \transl \jour Theory Probab. Appl. \yr 2020 \vol 65 \issue 3 \pages 359--374 \crossref{https://doi.org/10.1137/S0040585X97T990009} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000587381700002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85096068024}