Strong transience for one-dimensional Markov chains with asymptotically zero drifts

Lo, Chak Hei; Menshikov, Mikhail V.; Wade, Andrew R.

Abstract

For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at x decays as 1 / x as x → ∞ , we quantify degree of transience via existence of moments for conditional return times and for last exit times, assuming increments are uniformly bounded. Our proof uses a Doob h -transform, for the transient process conditioned to return, and we show that the conditioned process is also of Lamperti type with appropriately transformed parameters. To do so, we obtain an asymptotic expansion for the ratio of two return probabilities, evaluated at two nearby starting points; a consequence of this is that the return probability for the transient Lamperti process is a regularly-varying function of the starting point.

Citation

Lo, C. H., Menshikov, M. V., & Wade, A. R. (2023). Strong transience for one-dimensional Markov chains with asymptotically zero drifts. Stochastic Processes and their Applications, 104260. https://doi.org/10.1016/j.spa.2023.104260

• Journal Article Type: Article
• Acceptance Date: Nov 6, 2023
• Online Publication Date: Nov 10, 2023
• Publication Date: Nov 10, 2023
• Deposit Date: Nov 8, 2023
• Publicly Available Date: Nov 14, 2023
• Journal: Stochastic Processes and their Applications
• Print ISSN: 0304-4149
• Electronic ISSN: 1879-209X
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Pages: 104260
• DOI: https://doi.org/10.1016/j.spa.2023.104260
• Public URL: https://durham-repository.worktribe.com/output/1901643