Question: 11. Consider a Markov transition function P on a compact state space X. Prove that the corresponding Markov chain has at least one stationary mea-

11. Consider a Markov transition function P on a compact state space X. Prove that the corresponding Markov chain has at least one stationary mea- sure. (Hint: Take an arbitrary initial measure / and define un = (P*)"u, n 2 0. Prove that the sequence of measures Un = (o + ... + Un-1) is weakly compact, and the limit of a subsequence is a stationary measure.)

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