Non-homogeneous random walks on a half strip with generalized Lamperti drifts.

Abstract

We study a Markov chain on Undefined control sequence \RP, where Undefined control sequence \RP is the non-negative real numbers and S is a finite set, in which when the Undefined control sequence \RP-coordinate is large, the S-coordinate of the process is approximately Markov with stationary distribution πi on S. If μi(x) is the mean drift of the Undefined control sequence \RP-coordinate of the process at Undefined control sequence \RP, we study the case where ∑iπiμi(x)→0, which is the critical regime for the recurrence-transience phase transition. If μi(x)→0 for all i, it is natural to study the \emph{Lamperti\/} case where μi(x)=O(1/x); in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If μi(x)→di for di≠0 for at least some i, then it is natural to study the \emph{generalized Lamperti\/} case where μi(x)=di+O(1/x). By exploiting a transformation which maps the generalized Lamperti case to the Lamperti case, we obtain a recurrence classification and existence of moments results for the former. The generalized Lamperti case is seen to be more subtle, as the recurrence classification depends on correlation terms between the two coordinates of the process.