The principal results of Resnick and Neuts (1970) and Resnick (1971) concerning limiting distributions for the maxima of a sequence of random variables defined on a Markov chain have been extended to denumerable Markov chains. These results apply a fortiori to Markov renewal processes. The method of proof is to show that limit distributions are independent of the initial distribution of the chain and then to apply known results for stationary processes.
"Limit Theorems for Extreme Values of Chain-Dependent Processes." Ann. Probab. 3 (5) 773 - 779, October, 1975. https://doi.org/10.1214/aop/1176996264