Vol. 15, No. 3, 1965

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Ratio limit theorems for Markov chains

Sidney Charles Port

Vol. 15 (1965), No. 3, 989–1017

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.

Mathematical Subject Classification
Primary: 60.65
Received: 15 September 1964
Published: 1 September 1965
Sidney Charles Port