With any Harris-recurrent Markov chain one can associate a sequence of random times at which the chain has the same distribution, and the chain can thereby be shown to be equivalent to one having a recurrence point. This idea makes available a regeneration scheme for such chains, which is exploited in this paper to prove the ergodic theorem for semi-Markov processes, and a renewal theorem for Markov chains on a general state space.
"Limit Theorems for Semi-Markov Processes and Renewal Theory for Markov Chains." Ann. Probab. 6 (5) 788 - 797, October, 1978. https://doi.org/10.1214/aop/1176995429