The Chapman-Kolmogorov equations are fundamental equations in the theory of Markov processes that relate the transition probabilities of a stochastic process over different time intervals. These equations provide a way to express the probability of transitioning from one state to another in a Markov chain over multiple steps, linking short-term and long-term behaviors of the process. They are crucial for understanding how probabilities evolve over time in systems modeled by Markov chains.
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