Let X = (Xn : n 2 0) be a Markov chain with state space S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and transition matrix 1 2 3 4 5 6 7 8 9 1 0 0 0 0.3 0 0 0.7 0 0 2 0 0 3 0.6 0 0 0 0.1 0 0 3 0 0.6 0.4 0 0 0 0 0 0 4 0 0 0 0.5 0.5 0 0 0 0 P = 5 0 0 0 0 0 1 0 0 0 6 0 0 0 1 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0.4 0.6 9 0 0 0 0 0 0 0 0.6 0.4 a.) Identify the communication classes (one way is to draw the transition probability graph of X ) Then identify which states are recurrent/transient and which states are periodic. b.) This Markov chain is not irreducible, and there exists multiple invariant distributions. Find any two of them. c.) Provide an initial condition that guarantees the existence of limiting distribution