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This lecture covers the convergence of Markov chains, focusing on the invariant distribution and the Law of Large Numbers. The instructor explains the concept of invariant distribution, the interpretation of the Law of Large Numbers in the context of Markov chains, and the proof of the Markov LLN. The lecture delves into the computation of mean rewards and the length of excursions in Markov chains, emphasizing the importance of understanding the convergence properties of these stochastic processes.