Markov Chain Ergodic Theorem

Question

Consider a discrete time Markov Chain on countable state space $X_{0},X_{1},\ldots$. Assume that the chain satisfies the Foster Lyapunov criteria, and since it is countable state space chain, we conclude that it is positive recurrent. Will it still be true that for a bounded function $f$ we have $\frac{1}{n}\sum_{i=0}^{n}f(X_{i})\to Ef(X)$, where the expectation on rhs is wrt the stationary distribution of the process.

Answer 1

Yes, when I am thinking about this I dont know why not. As long as is ensured that every state can be expected to be returned to. Then all states in such an irreducible Markov chain being ergodic; follows the chain to be ergodic. With criteria of Foster Lyapunov fulfilled I shall understand that stability is ensured.

Of course, we are talking about a first order Markov chain of no memory.