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Non-homogeneous random walks on a semi-infinite strip.

Georgiou, Nicholas and Wade, Andrew R. (2014) 'Non-homogeneous random walks on a semi-infinite strip.', Stochastic processes and their applications., 124 (10). pp. 3179-3205.

Abstract

We study the asymptotic behaviour of Markov chains (Xn,ηn) on Z+×S, where Z+ is the non-negative integers and S is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of Xn, and that, roughly speaking, ηn is close to being Markov when Xn is large. This departure from much of the literature, which assumes that ηn is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for Xn given ηn. We give a recurrence classification in terms of increment moment parameters for Xn and the stationary distribution for the large- X limit of ηn. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between Xn (rescaled) and ηn. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on Z+ (the case where S is a singleton). Motivation arises from modulated queues or processes with hidden variables where ηn tracks an internal state of the system.

Item Type:Article
Keywords:Non-homogeneous random walk, Recurrence classification, Weak limit theorem, Lamperti’s problem, Modulated queues, Correlated random walk.
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1016/j.spa.2014.05.005
Publisher statement:© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
Date accepted:16 May 2014
Date deposited:16 June 2014
Date of first online publication:21 May 2014
Date first made open access:No date available

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