=+35.11. Let X1, X2 ,... be a Markov chain with countable state space S and transition probabilities p .,. A function , on S is excessive or superharmonic if (i) ≥

Ejpijp(j). Show by martingale theory that ( X) converges with probability 1 if p is bounded and excessive. Deduce from this that if the chain is irreducible and persistent, then « must be constant. Compare Problem 8.34.