Ratio limit theorems for Markov chains

In an irreducible, recurrent, Markov chain, with integer states, let N n (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(N n (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.Throughout this paper {X n } will be an irreducible, recurrent Markov chain with states in a denumerable set E and with nth step transition probabilities P* y .For convenience we may take E to be the integers.For discrete time Markov processes, ratio limits for the quantities P x (N n (A) = k) were first investigated by Kac [4] for certain special cases of partial sums of independent random variables with a common distribution.Recently these quantities have been intensively studied by Kesten and Spitzer [9] for the irreducible chains formed by the successive partial sums of independent, identically distributed, integerlattice-valued random vectors in r dimensions.They show the remarkable fact that in all such chains, for any two states x, y, any integer k ^ 0, and any finite nonempty set A, the limits exist and they explicitly find their values.Now, in general, limits (1.1) exist in very few recurrent chains and we shall have to be content with much weaker types of ratio limits if we want results of any generality.The weakest form of these ratio limits asserts that for any two states x, y, and any finite nonempty set A,exists for all k ^ 0. Although limits (1.2) exist in every positiverecurrent chain, there are null-recurrent chains in which these weak