Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355--386]. Compactness of the state space and a certain continuity condition allows us to simplify Kesten's proof considerably.
"Renewal theory for functionals of a Markov chain with compact state space." Ann. Probab. 31 (4) 2270 - 2300, October 2003. https://doi.org/10.1214/aop/1068646385