Nonhomogeneous random walks on a semiinfinite strip
Abstract
We study the asymptotic behaviour of Markov chains $(X_n,\eta_n)$ on $\mathbb{Z}_+ \times S$, where $\mathbb{Z}_+$ is the nonnegative 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 $X_n$ (rescaled) and $\eta_n$. Our results can be seen as generalizations of Lamperti's results for nonhomogeneous random walks on $\mathbb{Z}_+$ (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.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.2558
 Bibcode:
 2014arXiv1402.2558G
 Keywords:

 Mathematics  Probability;
 60J10 (Primary);
 60F05;
 60F15;
 60K15;
 60K25 (Secondary)
 EPrint:
 27 pages