Renewal approximation for the absorption time of a decreasing Markov chain

Abstract

We consider a Markov chain (M_n)_n≥0 on the set ℕ₀ of nonnegative integers which is eventually decreasing, i.e. ℙ{M_n+1<M_n | M_n≥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: M_k<a} under ℙ_n≔ℙ (· | M₀=n) as n→∞. Assuming that the decrements of (M_n)_n≥0 given M₀=n possess a kind of stationarity for large n, we derive sufficient conditions for the convergence in the minimal L^p-distance of ℙ_n(T−a_n)∕bn∈·) to some nondegenerate, proper law and give an explicit form of the constants a_n and b_n.