A transition probability matrix P is said to be doubly stochastic if the sum over each column

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A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one; that is,image text in transcribed

If such a chain is irreducible and consists of M + 1 states 0, 1, . . . , M, show that the long-run proportions are given byimage text in transcribed

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