Some dichotomy results for functionals of Harris recurrent Markov chains

Let {Xn}n⩾0 be a Harris recurrent Markov chain with state space E and let ξ be a measurable map from E to a separable Banach space B and setSn=∑k=1nξ(Xk), n=1,2,….Some integrabilities and tail behaviors of Sn over one excursion between two visits to a subset A of E are considered. It is shown that they hold either for all A in certain classes or for none in these classes.