1. Problem 15.36 Let {rj, j ∈ I} be a given probability distribution, where I is a finite set with N elements. Define the Markov chain {Xn} on the state space I by the one-step transition probabilities pij = 1 N−1 min(rj ri

, 1) for j= i and 414 Markov chains pii = 1− j:j=i pij.† Prove that {rj} is the equilibrium distribution of the process {Xn}. Hint: verify the reversibility property rj pjk = rk pkj for all j,k, which implies that rj = k∈I rkpkj for all j.