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 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

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

Bibliographic databases:

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

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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. V. I. Afanasyev, “Conditional limit theorem for maximum of random walk in a random environment”, Theory Probab. Appl., 58:4 (2014), 525–545
3. 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
4. V. I. Afanasyev, “On the non-recurrent random walk in a random environment”, Discrete Math. Appl., 28:3 (2018), 139–156
5. 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
6. V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Theory Probab. Appl., 63:3 (2019), 339–350
7. “Abstracts of talks given at the 3rd International Conference on Stochastic Methods”, Theory Probab. Appl., 64:1 (2019), 124–169
8. 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
9. 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
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