8. Let (X,,) be a Markov chain with transition matrix P and finite state space S. Suppose that a function f on 5S satisfies the condition /j-9 P(i,j)f(9) = ef (i) for some constant c # 0. (a) Find a sequence of constants , such that c,f (X,) is a martingale with respect to (X,,). (b) Show that if P is irreducible and c = 1 and the function f is constant. [Hint: consider the mean and mean square of f (.X,,) for a stationary chain (X,,).| (c) Give an example of a non-constant f satisfying the condition with e = 1 for a p t,q | walk on S$ := {0,1,...,N} with absorption at the boundary states 0 and N, and p # g