How often does a Harris recurrent Markov chain recur?

Abstract

Let {Xn}n≥0 be a Harris recurrent Markov chain with state space (E, ℰ), transition probability P(x, A) and invariant measure π. Given a nonnegative π-integrable function f on E, the exact asymptotic order is given for the additive functionals ∑nk=1 f(Xk), n = 1, 2, . . . in the forms of both weak and strong convergences. In particular, the frequency of {Xn}n>0 visiting a given set A ∈ ℰ with 0 < π(A) < +∞ is determined by taking f = IA. Under the regularity assumption, the limits in our theorems are identified. The one- and two-dimensional random walks are taken as the examples of applications.