Renewal theory for transient markov chains with asymptotically zero drift. / Denisov, Denis; Korshunov, Dmitry; Wachtel, Vitali.
In: Transactions of the American Mathematical Society, Vol. 373, No. 10, 10.2020, p. 7253-7286.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Renewal theory for transient markov chains with asymptotically zero drift
AU - Denisov, Denis
AU - Korshunov, Dmitry
AU - Wachtel, Vitali
N1 - Publisher Copyright: © 2020 American Mathematical Society Copyright: Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020/10
Y1 - 2020/10
N2 - We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain Xn in R, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by Xn 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/xα for some α ∈ (0, 1). The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case Xn2 /n converges weakly to a Γ-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for Xn1+α/n and further normal approximation is available.
AB - We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain Xn in R, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by Xn 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/xα for some α ∈ (0, 1). The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case Xn2 /n converges weakly to a Γ-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for Xn1+α/n and further normal approximation is available.
KW - Green function
KW - Lamperti's problem
KW - Renewal kernel
KW - Renewal measure
KW - Transient Markov chain
KW - LOCAL LIMIT-THEOREMS
KW - renewal measure
KW - renewal kernel
KW - FUNCTIONALS
UR - http://www.scopus.com/inward/record.url?scp=85092349569&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/df364e2b-8942-33d6-bef1-f9516c1b6de8/
U2 - 10.1090/tran/8167
DO - 10.1090/tran/8167
M3 - Article
AN - SCOPUS:85092349569
VL - 373
SP - 7253
EP - 7286
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 10
ER -
ID: 27417447