## The Annals of Mathematical Statistics

### 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.

#### Article information

Source
Ann. Math. Statist., Volume 37, Number 6 (1966), 1439-1462.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177699138

Digital Object Identifier
doi:10.1214/aoms/1177699138

Mathematical Reviews number (MathSciNet)
MR203811

Zentralblatt MATH identifier
0154.42901

JSTOR