Clustering of extreme events created by multiple correlated maxima

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

Dynamical counterexamples regarding the extremal index and the mean of the limiting cluster size distribution

“…This formula for the finite time cluster size distribution was used first in [16] and explicitly written for the first time in [4]. It appeared subsequently in [5,6]. This formula was derived during the proof of the convergence of REPP, which was based on a blocking type of argument.…”

“…Moreover, for sufficiently regular systems, a full dichotomy exists (see [4, 22]), that is, either is periodic and we have clustering or is non‐periodic and we have the absence of clustering with and a standard Poisson process as a limit for the REPP. In [5], the authors introduced a new device to create clustering: instead of considering observables maximised at a single point, they consider multiple maximising points and show that if these points are related by belonging to the same orbit, then a fake periodic behaviour emerges, which is responsible for the appearance of clustering of extreme observations. In this case, the maximal points need not to be periodic but the maximal set that they form, that is, the set of points where the observable attains the global maximum of the observable , is periodic in the sense that it recurs to itself after a finite number of iterations.…”

Extreme Value Laws for Dynamical Systems with Countable Extremal Sets

“…. , ξ k ∈ X , with k ∈ N, as in [HNT12,AFFR16], we assume that the maximum is achieved on a countable set M = {ξ i } i∈N 0 , which is the closure of a subset of the orbit of some chosen point ζ ∈ X . More precisely, for a certain point ζ ∈ X , we have that M := {f m i (ζ) : i ∈ N}.…”

“…Very recently, in [AFFR16], the authors considered the possibility of having a finite number of maximal points of ϕ. Moreover, it was shown that when these maximal points are correlated, in the sense of belonging to the same orbit of some point (not necessarily periodic), then clustering of exceedances is created by what turned out to be a mechanism that emulates some sort of fake periodic effect.…”

“…Namely, we will consider countably many maximal points, which, in compact phase spaces as we consider here, have accumulation points which complicate the analysis. The tools used in [AFFR16] were originally developed in [FFT12] and later refined in [FFT15]. They are based on some conditions on the dependence structure of the stochastic processes.…”