The Markov Renewal Theorem and Related Results
1997, v.3, Issue 1, 103-127
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 [G. Alsmeyer, On the Markov renewal theorem, Stoch. Process. Appl., 1994, v. 50, 37-56] 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.
Keywords: Markov random walks,ladder variables,Harris recurrence,regeneration epochs,coupling,cyclic decomposition