On the Average Length of Visits to an Ergodic Set

Let X1,X2, . . . be a sequence of random variables which are stationary and ergodic (see Breiman [1968, p. 109]) with values in a set S. Let μk be the stationary distribution for a sequence of length k so that for any fixed collection A1, A2, . . . , Ak of subsets of s, μk(A1, A2, . . . , Ak) = P[Xn ∈ A1,Xn+1 ∈ A2, . . . ,Xn+k−1 ∈ Ak] is independent of n, n = 1, 2, . . . . Now let B be a subset of S with 0 < μ1(B) < 1. If Xn ∈ B′, the complement of B in S, and for some v ≥ 1, Xn+1,Xn+2, . . . ,Xn+v are all in B while Xn+v+1 is in B′, we have a visit to B of length v. (Note that with this definition, a sequence which starts in B is not considered as a visit.) Let v1, v2, . . . , vj be the lengths of the first j visits to B. This note discusses the fact that as j → ∞, 1 j j ∑