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Non-homogeneous random walks on a semi-infinite strip

Georgiou, Nicholas; Wade, Andrew R.

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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.

Citation

Georgiou, N., & Wade, A. R. (2014). Non-homogeneous random walks on a semi-infinite strip. Stochastic Processes and their Applications, 124(10), 3179-3205. https://doi.org/10.1016/j.spa.2014.05.005

Journal Article Type Article
Acceptance Date May 16, 2014
Online Publication Date May 21, 2014
Publication Date Oct 1, 2014
Deposit Date Jan 22, 2014
Publicly Available Date Mar 29, 2024
Journal Stochastic Processes and their Applications
Print ISSN 0304-4149
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 124
Issue 10
Pages 3179-3205
DOI https://doi.org/10.1016/j.spa.2014.05.005
Keywords Non-homogeneous random walk, Recurrence classification, Weak limit theorem, Lamperti’s problem, Modulated queues, Correlated random walk.

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