Exercise 10.3 (ChapmanKolmogorov Equation) For any Markov chain {Xn}, denote its transition probabilities by pij(m, n). Prove

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Exercise 10.3 (Chapmanā€“Kolmogorov Equation) For any Markov chain {Xn}, denote its transition probabilities by pij(m, n). Prove that

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This identity can be compactly written in matrix form as

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