It follows from Theorem 4.2 that for a time reversible Markov chain Pi j Pjk Pki = Pik Pkj Pji, for all i, j, k It turns out that if the state space is finite and Pi j > 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

πj Pjk = πk Pkj are satisfied by πj = cPi j /Pji , where c is chosen so that 

j πj = 1.