Suppose Xn and Xn are independent ergodic Markov chains on the spaces S and Swith transition probabilities p ij and ij and stationary distributions π and π , respectively. Show that Zn = ( Xn , X n)...

Suppose
_Xn
and Xn are independent ergodic Markov chains on the spaces S and Swith transition probabilities p_ij
and ij and stationary distributions π and π , respectively. Show that Zn = (_Xn, X n) is an ergodic Markov chain on S × S with transition probabilities pij,_k
= pik_pj, and its stationary distribution is π_ij
= π_iπ
j