## The Annals of Mathematical Statistics

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

### The Existence and Uniqueness of Stationary Measures for Markov Renewal Processes

Ronald Pyke and Ronald Schaufele

#### 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

**Permanent link to this document**

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**

links.jstor.org

#### Citation

Pyke, Ronald; Schaufele, Ronald. The Existence and Uniqueness of Stationary Measures for Markov Renewal Processes. Ann. Math. Statist. 37 (1966), no. 6, 1439--1462. doi:10.1214/aoms/1177699138. https://projecteuclid.org/euclid.aoms/1177699138