For an irreducible positive recurrent Markov chain with stationary distribution , the expected return time to state is . More generally, expected occupation rewards in a return cycle equal their stationary rate times the expected cycle length.
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Kac's lemma, named after mathematician Mark Kac, is a result in probability theory concerning the expected value of a function of a random variable. It is particularly useful in the context of stochastic processes and the study of Brownian motion.