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July 2000 Markov additive processes and Perron-Frobenius eigenvalue inequalities
Colm O'Cinneide
Ann. Probab. 28(3): 1230-1258 (July 2000). DOI: 10.1214/aop/1019160333


We present a method for proving Perron –Frobenius eigenvalue inequalities. The method is to apply Jensen’s inequality to the change in a “random evolution ”over a regenerative cycle of the underlying finite-state Markov chain. One of the primary benefits of the method is that it readily gives necessary and sufficient conditions for strict inequality. It also gives insights into some of the conjectures of J.E.Cohen. Ney and Nummelin’s “Hypothesis 2” arises here as a condition for strict inequality, and we explore its ramifications in detail for a special family of Markov additive processes which we call “fluid models.” This leads to a connection between Hypothesis 2 and the condition “$P^T P$ irreducible” which arose in the work of Cohen, Friedland, Kato and Kelly.


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Colm O'Cinneide. "Markov additive processes and Perron-Frobenius eigenvalue inequalities." Ann. Probab. 28 (3) 1230 - 1258, July 2000.


Published: July 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1025.15027
MathSciNet: MR1797311
Digital Object Identifier: 10.1214/aop/1019160333

Primary: 60I10
Secondary: 15A42 , 16J27

Keywords: finite-state Markov chains , Markov additive processes , Perron–Frobenius eigenvalue inequalities

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 3 • July 2000
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