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Renewal approximation for the absorption time of a decreasing Markov chain

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  24 October 2016

Gerold Alsmeyer  and
Alexander Marynych
Gerold Alsmeyer*
Affiliation:
University of Münster
Alexander Marynych*
Affiliation:
Taras Shevchenko National University of Kyiv
*
* Postal address: Institute of Mathematical Statistics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149 Münster, Germany. Email address: gerolda@math.uni-muenster.de
** Postal address: Faculty of Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska 60, 01601 Kyiv, Ukraine. Email address: marynych@unicyb.kiev.ua
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Abstract

We consider a Markov chain (Mn)n≥0 on the set ℕ0 of nonnegative integers which is eventually decreasing, i.e. ℙ{Mn+1<Mn  |  Mn≥a}=1 for some a∈ℕ and all n≥0. We are interested in the asymptotic behavior of the law of the stopping time T=T(a)≔inf{k∈ℕ0:  Mk<a} under ℙn≔ℙ (·  |  M0=n) as n→∞. Assuming that the decrements of (Mn)n≥0 given M0=n possess a kind of stationarity for large n, we derive sufficient conditions for the convergence in the minimal Lp-distance of ℙn(T−an)∕bn∈·) to some nondegenerate, proper law and give an explicit form of the constants an and bn.

Keywords

Markov chainabsorption timeminimal Lp-distancerandom recursionrenewal theory

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 765 - 782
Copyright
Copyright © Applied Probability Trust 2016 

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