A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one; that is, i Pi j = 1, for all j If such a chain is irreducible and aperiodic and consists of M+1 states 0, 1,..., M, show that the long-run proportions are given by

πj = 1 M + 1

, j = 0, 1,..., M