2. Let X = (an 2 0) and Y = (Yn, n 2 0) be independent, irreducible aperiodic Markov chains. Say that the state space of X is V and the state space of Y is W. The state spaces may be of innite size. (a) (3 marks) If X is recurrent and Y is recurrent, does it follow that the product chain ((XmYn), n 2 0) is recurrent? (b) (3 marks) If X is recurrent and Y is transient, does it follow that the product chain ((XmYn), n 2 0) is recurrent? (c) (3 marks) If X is transient and Y is transient, does it follow that the product chain ((Xn, Y\"), n 2 0) is transient? (d) (3 marks) If X is recurrent and the state space W of Y is - nite, does it follow that the product chain ((Xn,Yn),n 2 0) is recurrent? Justify all your answers