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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Renewal theory for transient Markov chains with asymptotically zero drift


Authors: Denis Denisov, Dmitry Korshunov and Vitali Wachtel
Journal: Trans. Amer. Math. Soc. 373 (2020), 7253-7286
MSC (2010): Primary 60K05; Secondary 60J05, 60G42
DOI: https://doi.org/10.1090/tran/8167
Published electronically: August 6, 2020
MathSciNet review: 4155207
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Abstract: We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain $ X_n$ in $ \mathbb{R}$, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by $ X_n$ in the interval $ (x,x+1]$ is roughly speaking the reciprocal of the drift and tends to infinity as $ x$ grows.

For the first time we present a general approach relying on a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale-type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as $ 1/x$ or much slower than that, say as $ 1/x^\alpha $ for some $ \alpha \in (0,1)$.

The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case $ X_n^2/n$ converges weakly to a $ \Gamma $-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for $ X_n^{1+\alpha }/n$ and further normal approximation is available.


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Additional Information

Denis Denisov
Affiliation: Department of Mathematics, University of Manchester, United Kingdom
MR Author ID: 678962
Email: denis.denisov@manchester.ac.uk

Dmitry Korshunov
Affiliation: Department of Mathematics and Statistics, Lancaster University, United Kingdom
MR Author ID: 323844
Email: d.korshunov@lancaster.ac.uk

Vitali Wachtel
Affiliation: Institute of Mathematics, University of Augsburg, Germany
MR Author ID: 668465
Email: vitali.wachtel@math.uni-augsburg.de

DOI: https://doi.org/10.1090/tran/8167
Keywords: Transient Markov chain, renewal kernel, renewal measure, Lamperti's problem, Green function
Received by editor(s): July 18, 2019
Received by editor(s) in revised form: February 21, 2020
Published electronically: August 6, 2020
Article copyright: © Copyright 2020 American Mathematical Society