Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains $(X_n,\eta_n)$ on $\mathbb{Z}_+ \times S$, where $\mathbb{Z}_+$ 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 $X_n$ (rescaled) and $\eta_n$. Our results can be seen as generalizations of Lamperti's results for non-homogeneous 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.