i?-THEORY FOR MARKOV CHAINS

If {Xn} is a discrete-time ^-irreducible Markov chain on a measure space (#*, 3?) [4; p. 4], with «-step transition probabilities P"(x, A), it has been shown in [5] that there exists a subset ^R of 3F with the property that, for every Ae^R and 0-almost all x € W, the power series J^P(x, A)z" have the same radius of convergence R. Moreover, there is a countable partition of 2£ all of whose elements belong to f̂R. If all the power series diverge for z = R, and {X,,} is aperiodic, then there is a second subset €L of $F such that for any A e #L, lim P"(x> A)R n = n(x, A) < oo n-»oo exists for almost all xe%. The state space % can again be countably partitioned into elements of #L. In this paper we assume that a topology exists on <X, and investigate continuity conditions on the transition probabilities of {X,,} which will ensure that compact elements of fF lie in either gR or #L. A condition sufficient for both these desirable attributes is given in §3, and in §4 we consider weakening this condition. Examples are given to show that under the weaker condition compact sets may or may not belong to €R or #L, and some auxiliary conditions on 2£ are found which make the weaker continuity conditions sufficient for compact sets to belong to ̂ R and # t .