Entry time statistics to different shrinking sets

TL;DR

This paper studies the entry times to shrinking sets in $\psi$-mixing dynamical systems, establishing conditions under which the scaled entry times converge in distribution to an exponential law.

Contribution

It provides new conditions on families of sets ensuring the exponential distribution limit for scaled entry times in $\psi$-mixing systems.

Findings

Conditions for exponential law convergence of entry times

Results applicable to a broad class of $\psi$-mixing systems

Extension of classical recurrence time results

Abstract

We consider $\psi$-mixing dynamical systems $(\mathcal{X},T,B,\mu)$ and we find conditions on families of sets $\{\mathcal{U}_n\subset \mathcal{X}:n\in\mathbb{N}\}$ so that $\mu(\mathcal{U}_n)\tau_n$ tends in law to an exponential random variable, where $\tau_n$ is the entry time to $\mathcal{U}_n.$