Journal of Applied Probability

Renewal approximation for the absorption time of a decreasing Markov chain

Gerold Alsmeyer and Alexander Marynych

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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.

Article information

J. Appl. Probab., Volume 53, Number 3 (2016), 765-782.

First available in Project Euclid: 13 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Markov chain absorption time minimal 𝐿ₚ-distance random recursion renewal theory


Alsmeyer, Gerold; Marynych, Alexander. Renewal approximation for the absorption time of a decreasing Markov chain. J. Appl. Probab. 53 (2016), no. 3, 765--782.

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