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 $b_{ n}$.
Categories: math.PR