Clustering of extreme events created by multiple correlated maxima

TL;DR

This paper investigates how multiple correlated maxima in dynamical systems lead to new clustering mechanisms in extreme events, affecting extreme value laws, extremal index, and rare event processes.

Contribution

It introduces a novel clustering mechanism caused by multiple correlated maxima and analyzes its effects on extreme value theory in dynamical systems.

Findings

Correlated maxima create new clustering patterns.

Existence of limiting Extreme Value Laws is affected.

Impact on Extremal Index and Rare Events Points Processes.

Abstract

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The novelty is that we will consider observables achieving a global maximum value (possible infinite) at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point. These multiple correlated maxima can be seen as a new mechanism creating clustering. We recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point. We will study this mechanism for creating clustering and will address the existence of limiting Extreme Value Laws, the repercussions on the value of the Extremal Index, the impact on the limit of Rare Events Points Processes, the influence on clustering patterns and the competition of domains of attraction. We also consider briefly and for comparison purposes multiple uncorrelated maxima. The systems considered include expanding maps of the interval such as Rychlik maps but also maps with an indifferent fixed point such as Manneville-Pommeau maps.