|
This article is cited in 11 scientific papers (total in 11 papers)
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.
Received: 01.06.2010 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
Linking options:
https://www.mathnet.ru/eng/tvp4471https://doi.org/10.4213/tvp4471 https://www.mathnet.ru/eng/tvp/v57/i4/p625
|
|