Strong Ratio Limit Property for P-recurrent Markov Chains

for some positive integer d, then SRLP holds. It is also shown that the inequality (1.2) is always satisfied in a reversible chain. When P = 1, the results given here reduce to those of Orey [7] for recurrent chains, while for P> 1 they imply SRLP for a class of transient chains. (The definition of SRLP given by Orey required that 7=1 and that r< be independent of i since he was dealing with recurrent chains. The definition given here is the more natural one for transient chains.) The proofs given here were suggested by those used by Orey in the recurrent case. Although the proof given for the lemma is shorter than Orey's, the fundamental difference here is an analytic proof of Theorem 1 to replace the previous probabilistic one.2