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Non-homogeneous random walks on a semi-infinite strip
Authors:Nicholas Georgiou  Andrew R. Wade
Affiliation:Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, UK
Abstract:We study the asymptotic behaviour of Markov chains (Xn,ηn)(Xn,ηn) on Z+×SZ+×S, where Z+Z+ is the non-negative integers and SS is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of XnXn, and that, roughly speaking, ηnηn is close to being Markov when XnXn is large. This departure from much of the literature, which assumes that ηnηn is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for XnXn given ηnηn. We give a recurrence classification in terms of increment moment parameters for XnXn and the stationary distribution for the large- XX limit of ηnηn. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between XnXn (rescaled) and ηnηn. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on Z+Z+ (the case where SS is a singleton). Motivation arises from modulated queues or processes with hidden variables where ηnηn tracks an internal state of the system.
Keywords:primary, 60J10   secondary, 60F05, 60F15, 60K15, 60K25
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