1. Let X = (Xn, n 2 0) be an irreducible Markov chain with state space V and transition matrix P = (Pu,v)u,veV. (a) (3 marks) Show that the process Y = ((Xn, Xn+1), n 2 0) is a Markov chain with state space S = {(u, v) E V X V : Pu,v > 0}. Specify its state space and transition matrix. (b) (3 marks) Is Y necessarily irreducible? (c) (3 marks) If X is aperiodic, does it follow that all communica- tion classes of Y are aperiodic? (d) (3 marks) If 7 is a stationary distribution for X, what is a stationary distribution y for Y? (e) (3 marks) If X is reversible, does it follow that Y is reversible? Justify all your answers