Strong Ratio Limit Theorems for Null Recurrent Markov Chains and renewal Processes.

Suppose that X sub n is a Harris recurrent Markov Chain on a general state space (S[italic],S). Previous results of Athreya - Ney and Nummelin have shown that X sub n may be assumed to have a recurrent atom DELTA epsilon S without any loss of generality. Let T sub DELTA represent the first hitting time of X sub n of DELTA. We are interested in finding conditions for X sub n to have the strong ratio limit property (SRP) u sub n+1 ~ u sub n,(1), where u sub n = P sub DELTA (X sub n = DELTA), and a sub n ~ b sub n means that lim a sub n / b sub n = 1. If X sub n is positive recurrent, then lim u sub n = 1/(E sub DELTA [T sub DELTA]) > 0, and so, the result is trivially true. When X sub n is null recurrent (E sub DELTA [T sub DELTA ] = infinity ), however, the ratio has the indeterminate form 0/0, and (1) need not be true. The primary interest in the SRP stems from the fact that when (1) is true, the long run transition probabilities P sub K (X sub n epsilon E) all approach 0 at the some rate for a wide variety of initial distributions K on (S[itialic],S) and E !subset S. In addition, since u sub n is the renewal measure associated with the distribution of T sub DELTA, when the SRP is true, (1) is a generalization of Blackwell's Renewal Theorem to renewal sequences with infinite mean. In this paper, the previously known sufficient conditions of Kingman-Orey and Kesten are weakened and are generalized to arbitrary states spaces. Two types of conditions are considered: (i) direct hypotheses on the transition kernel of X sub n itself, and (ii) smoothness conditions on the distribution of T sub DELTA.