How Often Does a Harris Recurrent Markov Chain Recur?

Abstract

Let ${X _n}_ {n\geq 0}$ be a Harris recurrent Markov chain with state space $(E,\mathscr{E})$, transition probability $P(x, A)$ and invariant measure $\pi$ . Given a nonnegative $\pi$-integrable function $f$ on $E$, the exact asymptotic order is given for the additive functionals

\sum_{k=1}^{n} f(X_k) \quad n=1,2,\dots

in the forms of both weak and strong convergences. In particular, the frequency of ${X_n}_{n \geq 0}$ visiting a given set $A\in \mathscr{E}$ with $0 < \pi (A) < + \infty$ is determined by taking $f = I_A$. 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.