Limit theorem for a random walk in a reversible random environment
type: Item
instance of: scholarly article
title: Limit theorem for a random walk in a reversible random environment
zbMATH Open document ID: 0711.60067
DOI: 10.1007/BF00971945
published in: Lithuanian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let (\(\Omega\),\({\mathcal F},\mu)\) be a probability space and \(P^{(\omega)}\) be some probability measure associated with a homogeneous Markov chain X(k), \(k=0,1,...\), with values in \(Z^ d\) (d\(\geq 1)\) and with random transition probabilities. Denote by \(E^{\omega}(\cdot)\) and \(<\cdot >\) the means under the measures \(P^{(\omega)}\) and \(\mu\), respectively. Under some conditions on ``reversibility'' and ``small fluctuations'' the convergence of the process \(X_{\alpha,N}=X(N\cdot)/N^{1/\alpha}\) in the \(\mu\)-mean to a stable process \(W_{\alpha}(\cdot)\) with index \(\alpha\in (1,2)\) is obtained, i.e. for any finite set of points \(\{t_ i\}\subset (0,\infty)\) and any continuous, finite function f: \[ <| E^{(\omega)}f(X_{\alpha,N}(t_ 1),...,X_{\alpha,N}(t_ n))-E f(W_{\alpha}(t_ 1),...,W_{\alpha}(t_ n))| >\to 0\text{ for } N\to \infty. \]
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4169795
zbMATH Keywords: transition probabilities
zbMATH Keywords: reversibility
zbMATH Keywords: small fluctuations
author: Arvydas Astrauskas
reviewed by: K. Kubilius
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4169795