𝜆-continuous Markov chains

Abstract

0. Introduction and summary. We call a Markov operator P which has a representation P(x,A) = fAp(x,y)A(dy) with p(x,y) bivariate measurable a A-continuous Markov operator. It is a special kind of A-measurable Markov operator of E. Hopf. If the state space is discrete, every Markov operator is A-continuous where A assigns measure 1 to every state. In ?1 various definitions and preliminaries are given. In ?11 the existence of invariant measures for a A-continuous conservative P is proved. It is shown that the space is decomposed into at most countably many indecomposable closed sets C1,C2,-.. For each Ci there is a a-finite invariant measure 1,t which is equivalent to A on Ci and vanishes outside Ci. Every invariant measure is shown to be of the form lii. In ?111 convergence properties of In pn(Z,j)/ p'(z, y) are studied. It is shown that for a conservative ergodic P the limit of the ratio is f (x)/f (y) where f is the derivative of the invariant measure with respect to A. All these theorems are well known for a discrete state space (cf. [2, 1.9]). ?IV treats laws of large numbers. The approach used here is similar to that of Harris and Robbins [7]. It contains generalizations of theorems of Chung for discrete state spaces (cf. [2, 1.15]). The theory of A-measurable Markov operators is extensively used here. ?VI is devoted to some new results on A-measurable Markov operators which are used in this paper. In ?V the theory of Martin boundaries is investigated. The kernel K(x, y) used here is