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Renewal approximation for the absorption time of a decreasing Markov chain. (English) Zbl 1351.60023

Summary: We consider a Markov chain \((M_{n})_{n\geq0}\) on the set \(\mathbb{N}_{0}\) of nonnegative integers which is eventually decreasing, i.e. \(\mathbb{P}\{M_{n+1}<M_{n}|M_{n}\geq a\}=1\) for some \(a\in\mathbb{N}\) and all \(n\geq0\). We are interested in the asymptotic behavior of the law of the stopping time \(T=T(a):=\inf\{k\in\mathbb{N}_{0}:M_{k}<a\}\) under \(\mathbb{P}_{n}:=\mathbb{P}(\cdot|M_{0}=n)\) as \(n\to\infty\). Assuming that the decrements of \((M_{n})_{n\geq0}\) given \(M_{0}=n\) possess a kind of stationarity for large \(n\), we derive sufficient conditions for the convergence in the minimal \(L^{p}\)-distance of \(\mathbb{P}_{n}((T_{n}-a_n)/b_n\in\cdot)\) to some nondegenerate, proper law and give an explicit form of the constants \(a_{n}\) and \(b_{n}\).

MSC:

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