Sharp error terms and neccessary conditions for exponential hitting times in mixing processes

Abstract

We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in $\alpha$-mixing processes with exponential decay, $\phi$-mixing processes with a summable function $\phi$ and for general $\psi$-mixing processes with a finite alphabet. In the first case the bound is uniform as a function of the measure of the event. In the last two cases the bound depends also on the time scale t. This allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time. A uniform bound is a consequence. We present an example that shows that this bound is sharp. We also prove that second moments are not necessary for having the exponential law. Moreover, we prove a necessary condition for having the exponential limit law.