The Markov renewal theorem and related results. (English) Zbl 0906.60052

Alsmeyer, G.

Markov Process. Relat. Fields 3, No. 1, 103-127 (1997).

Summary: 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 by the author [Stochastic Processes Appl. 50, No. 1, 37-56 (1994; Zbl 0789.60066)] 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.

MSC:

60J05	Discrete-time Markov processes on general state spaces
60G50	Sums of independent random variables; random walks
60K05	Renewal theory
60K15	Markov renewal processes, semi-Markov processes

Keywords:

Markov random walks; ladder variables; Harris recurrence; regeneration epochs; coupling; cyclic decomposition