prove ergodic theorem for finite irreducible, aperiodic Markov Chain

Question

State and prove ergodic theorem for finite irreducible, aperiodic Markov Chain with transition probability matrix $P=(p_{ij})$.

I know what irreducible, aperiodic means. But I do not understand about what theorem it is referring. More precisely, what are the things I need to prove.

I know that if a finite markov chain is irreducible, it is called ergodic markov chain. But what to prove here if that is a definition. So what to prove here?

Thanks for any help.

Answer 1

Lemma C.1 (page 392) in the book "Markov Chains and Mixing Times (2nd edition)" written by David A.Levin and Yuval Peres contains the proof you want. (The key is to use the strong law of large numbers).