Renewal approximation for the absorption time of a decreasing Markov chain

TL;DR

This paper studies the asymptotic distribution of the absorption time of a decreasing Markov chain, providing conditions for convergence to a limiting law and explicit normalization constants.

Contribution

It introduces a renewal approximation approach for the absorption time of decreasing Markov chains and derives explicit conditions and constants for convergence.

Findings

Established conditions for convergence in distribution of normalized absorption times.

Derived explicit formulas for the normalization constants.

Provided a framework for analyzing decreasing Markov chains' absorption times.

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 $b_{n}$.