The Key Renewal Theorem for a Transient Markov Chain

We consider a time-homogeneous Markov chain $X_n$, $n\ge 0$, valued in ${\bf R}$\mathrm{.}Supposethatthischainistransient,thatis,$X_n$generatesa$\sigma$-finite renewal measure. We prove the key renewal theorem under condition that this chain has asymptotically homogeneous at infinity jumps and asymptotically positive drift.