Ratio limit theorems for markov chains

Abstract

In an irreducible, recurrent, Markov chain, with integer states, let Nn(A) be the occupation time of A by time n, where A is a finite set of states. Our principal concern in this paper is to investigate various “ratio limit theorem” for Px(Nn(A) = k). Criteria are given for various ratio limits to exist. The limits (when they exist) are shown to be expressible in terms of an integral over the set of states E completed with its dual recurrent boundary B ^. Applications are given to several specific Markov chains. © 1965 by Pacific Journal of Mathematics.