48. It follows from Theorem 4.2 that for a time reversible Markov chain PijPjkPki = PikPkjPji, for all ij, k It turns out that if the state space is finite and Pu > 0 for all i9j9 then the preceding is also a sufficient condition for time reversibility. (That is, in this case, we need only check Equation (4.21) for paths from / to / that have only two intermediate states.) Prove this.

Hint: Fix / and show that the equations njPjk = nkPkj are satisfied by ð, = cP^/Pß, where c is chosen so that £7 ð, = 1.