Explain intuitively why we require a Markov chain to be aperiodic in order for lim,-+0 Pi(Xn = j) = nj lim,-P(X, =/) = *; to hold? (That is, why is aperiodic a requirement for Theorem 8.4.8.) Theorem 8.4.8. If a Markov chain is irreducible and aperiodic with stationary distribution , then for all states i and j lim Pi(Xn = j) = *j. (8.21)