Coupling of Markov Chains: Ergodic Theorem

This lecture covers the concept of coupling of Markov chains, where two chains with the same state space and transition matrix are defined. The instructor explains how the distributions of these chains are related and how they evolve together. The lecture then delves into the proof of the ergodic theorem, focusing on the convergence properties of a Markov chain to a stationary distribution. Various lemmas and proofs are presented to demonstrate the convergence of the chain. The lecture concludes by establishing the equivalence between the ergodic theorem and the coupling of Markov chains, emphasizing the importance of understanding the coupling time and distribution convergence.