Question: An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector w). The mean time until the next occurrence of state si is

An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector w). The mean time until the next occurrence of state si is m¯i = Σk wkmki + wiri. Show that m¯i = zii/wi, by using the facts that wZ = w and mki = (zii − zki)/wi.

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