Derivation — Existence and Uniqueness of the Stationary Distribution

The previous lesson stated, without proof, that an irreducible recurrent chain has a stationary distribution with $\pi(y)=1/\mathbb{E}_y[T_y]$. This note supplies that proof via Durrett's "cycle trick" — build a candidate stationary measure by counting expected visits to each state during one excursion away from a fixed reference state, then show it is automatically balanced by the Markov property. The argument is a genuine piece of machinery (not just algebra), and it explains why the mean-return-time formula is true rather than merely asserting it.

Learning objectives. By the end of this note you should be able to: