71. It follows from Theorem 4.2 that for a time reversible Markov chain PijPjkPki = PikPkjPji ,...

Question:

71. It follows from Theorem 4.2 that for a time reversible Markov chain PijPjkPki = PikPkjPji , for all i, j, k It turns out that if the state space is finite and Pij > 0 for all i, j, then the preceding is also a sufficient condition for time reversibility. (That is, in this case, we need only check Equation (4.26) for paths from i to i that have only two intermediate states.) Prove this.

Hint: Fix i and show that the equations