Non-homogeneous random walks on a semi-infinite strip

Georgiou, Nicholas; Wade, Andrew R.

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: Jun 16, 2014
• Journal: Stochastic Processes and their Applications
• Print ISSN: 0304-4149
• Electronic ISSN: 1879-209X
• 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.
• Public URL: https://durham-repository.worktribe.com/output/1444926