Gnedenko's work (1943) showing the existence of a limiting distribution for the maximum of the first $n$ terms in a sequence of suitably normalized independent identically distributed random variables is extended to sequences of random variables defined on a positive-recurrent denumerable Markov chain.
"The Limiting Distribution of Maxima of Random Variables Defined on a Denumerable Markov Chain." Ann. Probab. 2 (1) 103 - 111, February, 1974. https://doi.org/10.1214/aop/1176996755