 Kac's lemma

For an irreducible positive recurrent discrete-time Markov chain with stationary distribution $\pi$, the expected return time to state $i$ is $1/{\pi }_i$. More generally, expected occupation rewards in a return cycle equal their stationary rate times the expected cycle length.

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.