Frequently visited sets for random walks

We study the occupation measure of various sets for a symmetric transient random walk in Z d with finite variances. Let μ n X ( A ) denote the occupation time of the set A up to time n. It is shown that sup x ∈ Z d μ n X ( x + A ) / log n tends to a finite limit as n → ∞ . The limit is expressed in...

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Bibliographic Details
Published in:Stochastic processes and their applications 2005-09, Vol.115 (9), p.1503-1517
Main Authors: Csáki, Endre, Földes, Antónia, Révész, Pál, Rosen, Jay, Shi, Zhan
Format: Article
Language:English
Subjects:
Exact sciences and technology
Limit theorems
Markov processes
Mathematics
Occupation measure
Probability
Probability and statistics
Probability theory and stochastic processes
Random walk
Random walk Occupation measure Strong theorems
Sciences and techniques of general use
Stochastic processes
Strong theorems
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Summary:We study the occupation measure of various sets for a symmetric transient random walk in Z d with finite variances. Let μ n X ( A ) denote the occupation time of the set A up to time n. It is shown that sup x ∈ Z d μ n X ( x + A ) / log n tends to a finite limit as n → ∞ . The limit is expressed in terms of the largest eigenvalue of a matrix involving the Green function of X restricted to the set A. Some examples are discussed and the connection to similar results for Brownian motion is given.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2005.04.003