Renewal approximation for the absorption time of a decreasing Markov chain

Abstract

We consider a Markov chain $(M_n{)}_{n\ge 0}$ on the set ${\mathbb{N}}_0$ of nonnegative integers which is eventually decreasing, i.e. $\mathbb{P}\{M_{n+1}<M_n|M_n\ge a\}=1$ for some $a\in \mathbb{N}$ and all $n\ge 0$. We are interested in the asymptotic behaviour 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 $(M_n{)}_{n\ge 0}$ given $M_0=n$ possess a kind of stationarity for large $n$, we derive sufficient conditions for the convergence in minimal $L^p$-distance of ${\mathbb{P}}_n((T-a_n)/b_n\in \cdot )$ to some non-degenerate, proper law and give an explicit form of the constants $a_n$ and $b_n$.