## Journal of Applied Probability

### Renewal approximation for the absorption time of a decreasing Markov chain

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 13 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1476370775

Mathematical Reviews number (MathSciNet)
MR3570093

Zentralblatt MATH identifier
1351.60023

#### Citation

Alsmeyer, Gerold; Marynych, Alexander. Renewal approximation for the absorption time of a decreasing Markov chain. J. Appl. Probab. 53 (2016), no. 3, 765--782. https://projecteuclid.org/euclid.jap/1476370775

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