Renewal approximation for the absorption time of a decreasing Markov chain

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 | Mn≥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: 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(T−an)∕bn∈·) to some nondegenerate, proper law and give an explicit form of the constants an and bn.