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December, 1966 The Existence and Uniqueness of Stationary Measures for Markov Renewal Processes
Ronald Pyke, Ronald Schaufele
Ann. Math. Statist. 37(6): 1439-1462 (December, 1966). DOI: 10.1214/aoms/1177699138


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.


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Ronald Pyke. Ronald Schaufele. "The Existence and Uniqueness of Stationary Measures for Markov Renewal Processes." Ann. Math. Statist. 37 (6) 1439 - 1462, December, 1966.


Published: December, 1966
First available in Project Euclid: 27 April 2007

zbMATH: 0154.42901
MathSciNet: MR203811
Digital Object Identifier: 10.1214/aoms/1177699138

Rights: Copyright © 1966 Institute of Mathematical Statistics

Vol.37 • No. 6 • December, 1966
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