The vector of state probabilities for any period is equal to the vector of state probabilities for the preceding period multiplied by the matrix of transition probabilities.

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This is the fundamental property of Markov chains, known as the Markov property. It states that the probability of transitioning from one state to another depends only on the current state and not on any historical information. This allows us to compute the probability distribution of future states based on the current state and the transition probabilities.

The Vector of State Probabilities for Any Period Is Equal