Show that an irreducible MC is aperiodic if there is at least one positive element on the main diagonal of its TPM. Hint. Let X be an irreducible MC with a TPM P = p² Pi ... Pii I where Pii > 0. Then, where på > (pii)² > 0 (explain why?). By induction, p>0, implying that the set of returns to state i counted in number of steps is not only countable but actually equals N. Therefore the GCD of all those return numbers equals 1 implying that state į is aperiodic. Because periodicity is a class-relation, we conclude that X is aperiodic. In fact, it turns out that p > 0 for all n.

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