It follows from Theorem 4.2 that for a time reversible Markov chain PyPjk PPPP, for all i, j, k It turns out that if the state space is finite and P > 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 and show that the equations xjPjk = k Pkj are satisfied by x, cP/P, where c is chosen so that ,, = 1.