Convergence for periodic Markov chains

Consider a discrete-time Markov chain (Xn : n > 0) ∼ Markov(λ, P ) with countable state space S. In the last lecture, we proved the following theorem for aperiodic Markov chains. Theorem (Aperiodic case). Assume P is irreducible and positive recurrent with unique invariant measure π. If, in addition, P is aperiodic, then P(Xn = j)→ πj as n→∞ for all j ∈ S. In particular, p (n) ij → πj as n→∞ for all i, j ∈ S.