The Key Renewal Theorem for a Transient Markov Chain

TL;DR

This paper proves a key renewal theorem for transient Markov chains on the real line, assuming asymptotic homogeneity and positive drift at infinity, extending renewal theory to this class of processes.

Contribution

It establishes the key renewal theorem for transient Markov chains with asymptotic homogeneity and positive drift, a significant extension of classical renewal theory.

Findings

Proves the key renewal theorem under specified conditions.

Extends renewal theory to transient Markov chains on ℝ.

Provides conditions for asymptotic homogeneity and positive drift.

Abstract

We consider a time-homogeneous Markov chain $X_n$, $n\ge0$, valued in ${\bf R}$. Suppose that this chain is transient, that is, $X_n$ generates a $\sigma$-finite renewal measure. We prove the key renewal theorem under condition that this chain has asymptotically homogeneous at infinity jumps and asymptotically positive drift.