Non-homogeneous random walks on a semi-infinite strip

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Georgiou, Nicholas¹  Wade, Andrew R.¹ 

[1] Univ Durham, Dept Math Sci, Durham DH1 3LE, England

关键词: Non-homogeneous random walk;    Recurrence classification;    Weak limit theorem;    Lamperti's problem;    Modulated queues;    Correlated random walk;   

DOI  :  10.1016/j.spa.2014.05.005

来源: Elsevier

【 摘 要 】

We study the asymptotic behaviour of Markov chains (X-n, eta(n)) on Z(+) x 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 X-n, and that, roughly speaking, eta(n) is close to being Markov when X-n is large. This departure from much of the literature, which assumes that eta(n) is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for X-n given eta(n). We give a recurrence classification in terms of increment moment parameters for X-n and the stationary distribution for the large-X limit of eta(n). In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between X-n (resealed) and eta(n). 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 eta(n) tracks an internal state of the system. (C) 2014 The Authors. Published by Elsevier B.V.

【 授权许可】

Free