In the following problem you are supposed to apply the positivity lemma. Let P be irreducible aperiodic finite transition matrix. Prove a) For any states i, j there exists an nij so that P() > 0 for all n ≥ Nij. b) For any states i, j there exists an no so that P(n) > 0 for all n ≥ no. c) There exists an no and a d> 0 so that Pn> 8 10 1' for n ≥ no. Here 1; = 1 for i ES.

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