Extremal Index, Hitting Time Statistics and periodicity

TL;DR

This paper develops conditions for the existence of an Extremal Index in stationary processes, linking it to periodic phenomena, and applies these ideas to dynamical systems and hitting time statistics.

Contribution

It introduces a new framework to identify the Extremal Index related to periodicity and applies it to dynamical systems and hitting time analysis.

Findings

Conditions established for Extremal Index existence

Application to dynamical systems with various measures

Analysis of Extreme Value Laws and hitting time statistics

Abstract

We give conditions to prove the existence of an Extremal Index for general stationary stochastic processes by detecting the presence of one or more underlying periodic phenomena. This theory, besides giving general useful tools to identify the extremal index, is also tailored to dynamical systems. In fact, we apply this idea to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. As in the authors' previous works on this topic, the analogy of these laws in the context of hitting time statistics is explained and exploited extensively.