=+5. Consider the Cartesian product state space A × B, where A = {0, 1,...,a − 1}, B = {0, 1,...,b − 1}, and a and b are positive integers. Define a Markov chain that moves from (x, y) to (x+1 mod

a, y) or (x, y+1 mod

b) with equal probability at each epoch. Show that the chain is irreducible. Also show that it is aperiodic if and only if the greatest common divisor of a and b is 1. (Hints: It helps to consider the special state (0, 0). See Proposition A.1.4 of Appendix A.1.)

7.9 Problems 175