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September 2016 Renewal approximation for the absorption time of a decreasing Markov chain
Gerold Alsmeyer, Alexander Marynych
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J. Appl. Probab. 53(3): 765-782 (September 2016).

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 | Mna}=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(Tan)∕bn∈·) to some nondegenerate, proper law and give an explicit form of the constants an and bn.

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Gerold Alsmeyer. Alexander Marynych. "Renewal approximation for the absorption time of a decreasing Markov chain." J. Appl. Probab. 53 (3) 765 - 782, September 2016.

Information

Published: September 2016
First available in Project Euclid: 13 October 2016

zbMATH: 1351.60023
MathSciNet: MR3570093

Subjects:
Primary: 60F05
Secondary: 60J10

Keywords: Absorption time , Markov chain , minimal 𝐿ₚ-distance , random recursion , renewal theory

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 3 • September 2016
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