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

• Type: Article in International Scientific Journal
• Date: 2020

Authors

Abadi, M (Author) Other; Ana Cristina Moreira Freitas (Author) FEP; Jorge Milhazes Freitas (Author) FCUP

Journal

Title: Journal of the London Mathematical Society

Serial No. 102 Vol. 102 No. 2

Pages: 670-694

ISSN: 0024-6107

Publisher: Wiley-Blackwell

Indexing

ISI Web of Knowledge - 9 Citations

Scopus - 8 Citations

Other information

Authenticus ID: P-00S-3KT

DOI: 10.1112/jlms.12332

Abstract (EN): The extremal index (EI) is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non-periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an EI equal to 0 with that of an EI larger than 0. Using bi-dimensional point processes, we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the cluster size distribution introduced earlier and prove that ergodicity is enough to establish that the finite versions of the reciprocal of the EI and of the mean of the cluster size distribution do coincide. © 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

Language: English

Type (Professor's evaluation): Scientific