Convergence to Equilibrium: The Coupling Argument and the Ergodic Theorem

Every earlier lesson in this topic has treated the stationary distribution $\pi$ as a fixed point of the transition matrix — a distribution that, once reached, is never left. This note proves the much stronger and more useful fact: for an irreducible, aperiodic chain with stationary distribution $\pi$, the $n$-step transition probabilities actually converge to $\pi$ as $n\to\infty$, starting from any initial state. The proof uses coupling — running two independent copies of the chain and showing they eventually meet — one of the most elegant and reusable techniques in all of probability. A companion result, the ergodic theorem, gives the same conclusion in a different, often more practically useful form: long-run time averages along a single realised path converge to averages against $\pi$, with no need to run the chain many times.

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