The classical Extreme Value Theory deals with independent random variables. If random variables are dependent, large values tend to cluster (that is, one large value is followed by a series of large values). It is of interest to describe probabilistically the clustering and estimate the relevant cluster functionals. We consider disjoint blocks, sliding blocks and runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain consistency results and central limit theorems under conditions that can be easily verified for a large class of short-range dependent models. In particular, we show that in the Peak-over-Threshold framework, all the estimators have the same limiting variances. This solves a longstanding open problem and is in contrast to the Block Maxima method. Our findings are illustrated by simulation experiments.

Cette thèse traite principalement des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d'extrêmes de séquences et champs aléatoires faiblement dépendants. Des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d'extrême de séries temporelles stationnaires sont donnés par Drees & Rootzén [2010] sous des conditions de régularité absolue (ou "ß-mélange"). Cependant, ces conditions de dépendance de type mélange sont très restrictives : elles sont particulièrement adaptées aux modèles dans la finance et dans l'histoire, et elles sont de plus compliquées à vérifier. Généralement, pour d'autres modèles fréquemment rencontré dans les domaines applicatifs, les conditions de mélange ne sont pas satisfaites. En revanche, les conditions de dépendance faible, selon Doukhan and Louhichi [1999] et Dedecker & Prieur [2004a], sont des conditions qui généralisent les notions de mélange et d'association. Elles sont plus simple à vérifier et peuvent être satisfaites pour de nombreux modèles. Plus précisément, sous des conditions faibles, tous les processus causals ou non causals sont faiblement dépendants: les processus Gaussien, associés, linéaires, ARCH(∞), bilinéaires et notamment Volterra entrent dans cette liste. À partir de ces conditions favorables, nous étendons certains des théorèmes limites de Drees & Rootzén [2010] à processus faiblement dépendants. En outre, comme application des théorèmes précédents, nous montrons la convergence en loi de l'estimateur de l'extremogramme de Davis & Mikosch [2009] et l'estimateur fonctionnel de l'indice extrémal de Drees [2011] sous dépendance faible. Nous démontrons un théorème de la valeur extrême pour les champs aléatoires stationnaires faiblement dépendants et nous proposons, sous les mêmes conditions, un critère du domaine d'attraction d'une loi d'extrêmes. Le document se conclue sur des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d’extrêmes de champs aléatoires stationnaires faiblement dépendants, et met en évidence la convergence en loi de l'estimateur d'un extremogramme de processus spatio-temporels stationnaires faiblement dépendants en tant qu'application. / This thesis deals mainly with limit theorems for empirical processes of extreme cluster functionals of weakly dependent random fields and sequences. Limit theorems for empirical processes of extreme cluster functionals of stationnary time series are given by Drees & Rootzén [2010] under absolute regularity (or "ß-mixing") conditions. However, these dependence conditions of mixing type are very restrictive: on the one hand, they are best suited for models in finance and history, and on the other hand, they are difficult to verify. Generally, for other models common in applications, the mixing conditions are not satisfied. In contrast, weak dependence conditions, as defined by Doukhan & Louhichi [1999] and Dedecker & Prieur [2004a], are dependence conditions which generalises the notions of mixing and association. These are easier to verify and applicable to a wide list of models. More precisely, under weak conditions, all the causal or non-causal processes are weakly dependent: Gaussian, associated, linear, ARCH(∞), bilinear and Volterra processes are some included in this list. Under these conveniences, we expand some of the limit theorems of Drees & Rootzén [2010] to weakly dependent processes. These latter results are used in order to show the convergence in distribution of the extremogram estimator of Davis & Mikosch [2009] and the functional estimator of the extremal index introduced by Drees [2011] under weak dependence. We prove an extreme value theorem for weakly dependent stationary random fields and we propose, under the same conditions, a domain of attraction criteria of a law of extremes. The document ends with limit theorems for the empirical process of extreme cluster functionals of stationary weakly dependent random fields, deriving also the convergence in distribution of the estimator of an extremogram for stationary weakly dependent space-time processes.

The thesis is focused on extreme value theory and its applications. Initially, extreme value distribution is introduced and its properties are discussed. At this basis are described two models mostly used for an extreme value analysis, i.e. the block maxima model and the Pareto-distribution threshold model. The first one takes advantage in its robustness, however recently the threshold model is mostly preferred. Although the threshold choice strongly affects estimation quality of the model, an optimal threshold selection still belongs to unsolved issues of this approach. Therefore, the thesis is focused on techniques for proper threshold identification, mainly on adaptive methods suitable for the use in practice. For this purpose a simulation study was performed and acquired knowledge was applied for analysis of precipitation records from South-Moravian region. Further on, the thesis also deals with extreme value estimation within a stationary series framework. Usually, an observed time series needs to be separated to obtain approximately independent observations. The use of the advanced theory for stationary series allows to avoid the entire separation procedure. In this context the commonly applied separation techniques turn out to be quite inappropriate in most cases and the estimates based on theory of stationary series are obtained with better precision.

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