The Existence and Uniqueness of Stationary Measures for Markov Renewal Processes

Abstract

In [4], Doob shows that $F^\ast(x) = \mu^{-1} \int^x_0 \lbrack 1 - F(u)\rbrack du$ is a stationary probability measure for a renewal process when the common distribution function $F$ has a finite mean $\mu$. In [2], Derman shows that an irreducible, null recurrent Markov chain (MC) has a unique positive stationary measure. In this paper, similar results are obtained for a class of irreducible recurrent Markov renewal processes (MRP). Since MRP's are generalizations of MC's and renewal processes these results generalize those mentioned above. Stationary measures are also derived for a class of MRP's with auxiliary paths.