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

Lo, Chak Hei; Wade, Andrew R.

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Authors

Chak Hei Lo



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.

Citation

Lo, C. H., & Wade, A. R. (2017). Non-homogeneous random walks on a half strip with generalized Lamperti drifts. Markov processes and related fields, 23(1), 125-146

Journal Article Type Article
Acceptance Date Jul 24, 2016
Publication Date Jan 1, 2017
Deposit Date Mar 11, 2016
Publicly Available Date Mar 28, 2024
Journal Markov processes and related fields.
Print ISSN 1024-2953
Publisher Polymat
Peer Reviewed Peer Reviewed
Volume 23
Issue 1
Pages 125-146
Publisher URL http://math-mprf.org/journal/articles/id1455/

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