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... Z is irreducible. Finally observe that T * c is the first return time of the coupled Markov chain to the state (x * , x * ). Since Z is irreducible and has a stationary distribution, it is positive recurrent according to Thm. 3.21. By Thm. 3.18, this implies P[T * c < ∞] = 1, which completes the proof of the statement. Fig. 9 summarizes the long run behavior of irreducible and aperiodic Markov ...
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