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.
"Markov additive processes and Perron-Frobenius eigenvalue inequalities." Ann. Probab. 28 (3) 1230 - 1258, July 2000. https://doi.org/10.1214/aop/1019160333