Theorem 5.5.11 — the stationary probability is the reciprocal mean return time

Let $p$ be irreducible with stationary distribution $\pi$. Then for every state $x$ the mean return time $\mathbb{E}_xT_x=\sum_{n\ge1}n\,f^n(x,x)$ is finite, and

This gives the stationary probability a second meaning, and a purely local one. As a fixed point of the transition matrix, $\pi(x)$ is a global object computed from the whole chain; as a reciprocal mean return time it is determined by the excursions from $x$ alone. Durrett's own remark — that the identity "seems incredible" — is a fair reaction, and the reason it holds is that the chain spends, on average, one visit per excursion, so the long-run fraction of time at $x$ is one over the mean excursion length.

The identity is the engine of renewal-reward arguments throughout applied probability: it converts a stationary probability, which is often hard to compute, into an expected return time, which is often easy, and conversely.

Formalization Note Finiteness of the mean return time is asserted explicitly, as summability of $n\,f^n(x,x)$, rather than left to a division convention: the mean return time is an unconditional sum, which would silently be zero for a divergent series, and $1/0$ would then read as the correct value for the wrong reason. The identity itself is stated as a product equal to one for the same reason. Irreducibility and the existence of a stationary distribution together force positive recurrence, which is what makes both halves true.