Hitting Times and Positions in Rare Events

TL;DR

This paper develops abstract limit theorems for rare events in ergodic systems, showing Poisson limits and iid positions without relying on classical mixing assumptions, and illustrates these results in simple systems.

Contribution

The paper introduces new limit theorems for rare events that do not depend on traditional mixing conditions, broadening understanding of their asymptotic behavior.

Findings

Poisson asymptotics for rare events in ergodic systems

Asymptotic iid distribution of positions within rare events

Applicability to simple prototypical systems

Abstract

We establish abstract limit theorems which provide sufficient conditions for a sequence $(A_{l})$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the $A_{l}$ to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to $A_{l}$ before some time $\tau_{l}$ which is of order $o(1/\mu(A_{l}))$. In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.