Fonctionnelles de clusters d’extrêmes de processus et champs aléatoires.
José Gomez Garcia
MIA-Paris, Agroparitsech
https://gomezgarcia.users.lmno.cnrs.fr/
Date(s) : 16/04/2021 iCal
14h00 - 16h00
Par ailleurs, comme une grande partie des estimateurs utilisés dans l’analyse statistique d’extrêmes peuvent être écrits en termes de fonctionnelles de clusters, nous utilisons ce fait et les résultats précédents pour fournir des théorèmes limites pour certains estimateurs comme l’estimateur de l’extrémogramme et l’indice extrémal sous de faibles conditions.
L’exposé se conclura avec des extensions aux champs aléatoires et des applications.
Functional clusters of extreme processes and random fields
Functionals of clusters of extremes were introduced and studied by Yun (2000) for d-order Markov chains. A few years later, Segers (2003) and Drees & Rootzén (2010) provide asymptotic results in the context of one-dimensional and multidimensional stationary time series respectively. However, these results are demonstrated under mixture type dependence conditions, which are very restrictive: they are particularly suitable for models in finance, and they are very complicated to handle mathematically. Generally, for other models frequently encountered in application domains, the mixing conditions are not satisfied. In contrast, the weak dependence conditions of Doukhan & Louhichi (1999) and Dedecker & Prieur (2004a) are more general and include a large list of models. Specifically, under weak conditions, all causal or non-causal processes are weakly dependent: Gaussian, associated, linear, ARCH(\infty), bilinear and notably Volterra processes fall into this list. From these favorable conditions, we extend some of these results from cluster functionals to the framework of weakly dependent processes.
Moreover, since a large part of the estimators used in statistical analysis of extremes can be written in terms of cluster functionals, we use this fact and the previous results to provide limit theorems for some estimators such as the extremogram estimator and the extremal index under weak conditions.
The talk will conclude with extensions to random fields and applications.
https://www.mdpi.com/2227-7390/9/3/212
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