An |S| x |S| transition matrix P is said to be doubly stochastic if SUM_xS P(x, y) = 1 for each y S.

Suppose that a Markov chain with doubly stochastic transition matrix P is irreducible and has state space S = {0, 1, . . . , M 1}. Verify that, for all y S, the equilibrium distribution satisfies (y) = 1/M.