Electronic Journal of Probability

Random walks in a moderately sparse random environment

Dariusz Buraczewski, Piotr Dyszewski, Alexander Iksanov, Alexander Marynych, and Alexander Roitershtein

Full-text: Open access

Abstract

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_{n})_{n\in \mathbb{N} \cup \{0\}}$ in a sparse random environment $(S_{k},\lambda _{k})_{k\in \mathbb{Z} }$ is a nearest neighbor random walk on $\mathbb{Z} $ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z} \setminus \{\ldots ,S_{-1},S_{0}=0,S_{1},\ldots \}$ and jumps to the right (left) with the random probability $\lambda _{k+1}$ ($1-\lambda _{k+1}$) from the point $S_{k}$, $k\in \mathbb{Z} $. Assuming that $(S_{k}-S_{k-1},\lambda _{k})_{k\in \mathbb{Z} }$ are independent copies of a random vector $(\xi ,\lambda )\in \mathbb{N} \times (0,1)$ and the mean $\mathbb{E} \xi $ is finite (moderate sparsity) we obtain stable limit laws for $X_{n}$, properly normalized and centered, as $n\to \infty $. While the case $\xi \leq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} \xi =\infty $ (strong sparsity) will be analyzed in a forthcoming paper.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 69, 44 pp.

Dates
Received: 27 April 2018
Accepted: 7 June 2019
First available in Project Euclid: 28 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1561687602

Digital Object Identifier
doi:10.1214/19-EJP330

Mathematical Reviews number (MathSciNet)
MR3978219

Zentralblatt MATH identifier
07089007

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
branching process in a random environment with immigration perpetuity random difference equation random walk in a random environment

Rights
Creative Commons Attribution 4.0 International License.

Citation

Buraczewski, Dariusz; Dyszewski, Piotr; Iksanov, Alexander; Marynych, Alexander; Roitershtein, Alexander. Random walks in a moderately sparse random environment. Electron. J. Probab. 24 (2019), paper no. 69, 44 pp. doi:10.1214/19-EJP330. https://projecteuclid.org/euclid.ejp/1561687602


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