Kac's lemma

Let $P$ be an irreducible Markov chain on a countable state space $V$ (nonnegative entries, rows summing to one as convergent series, every state reaching every other), and suppose $\pi$ is a stationary distribution for it: nonnegative, summing to one, with $\sum_x\pi(x)P(x,y)=\pi(y)$ for every $y$. For a nonempty set of states $S$ and $x\in S$, the first return time to $S$ is $\tau^+_S=\min\{t\ge1:X_t\in S\}$, with expectation given by the tail-sum formula $\mathbb E_x(\tau^+_S)=\sum_{t\ge0}\mathbb P_x\{\tau^+_S>t\}$.

The theorem (Kac's lemma, Lemma 21.13 of Levin–Peres–Wilmer) asserts the exact identity

the series over $S$ converging to exactly $1$.

Started from stationarity conditioned on being in $S$, the expected time to return to $S$ is exactly $1/\pi(S)$ — and for a singleton, $\mathbb E_x(\tau^+_x)=1/\pi(x)$: stationary mass is inverse return time. The identity is a mass-transport double count — every time-step of the stationary chain is the $k$-th step of exactly one excursion from $S$ — and it is the quantitative link between stationary distributions and return times that drives the positive-recurrence theory of this mission.