On the Distribution of First Passage and Return Times for Small Sets

Abstract

For a Harris recurrent Markov chain with invariant initial distribution $\pi$, we consider the return times $\tau_\varepsilon$ to state sets $A_\varepsilon$ with $0 < \pi(A_\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$ and show that, provided the probability of early returns to $A_\varepsilon$ approaches 0, the $\tau_\varepsilon$, multiplied by suitable scaling factors, are asymptotically exponentially distributed.