The Key Renewal Theorem for a Transient Markov Chain

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.