The Markov Renewal Theorem and Related Results

We give a new probabilistic proof of the Markov renewal theorem for Markov random walks with positive drift and Harris recurrent driving chain. It forms an alternative to the one recently given in [1] and follows more closely the probabilistic proofs provided for Blackwell’s theorem in the literature by making use of ladder variables, the stationary Markov delay distribution and a coupling argument. A major advantage is that the arguments can be refined to yield convergence rate results.