Suppose C is a closed recurrent communicating class for a Markov chain, if we restrict the chain on this class, the result is still a Markov chain with state space C. The resulting Markov chain is irreducible and recurrent. In this lecture, we will focus on the long-time behavior of irreducible and recurrent Markov chain. First, we have the following related definition. Definition 10.1. For a M...