The tail dependence coefficient (TDC) is a natural tool to describe extremal dependence. Estimation of the tail dependence coefficient can be performed via empirical process theory. In case of extremal independence, the limit degenerates and hence one cannot construct a test for extremal independence. In order to deal with this issue, we consider an analog of the covariance matrix, namely the extremogram matrix, whose entries depend only on extremal observations. We show that under the null hypothesis of extremal independence and for finite dimension d ≥ 2, the largest eigenvalue of the sample extremogram matrix converges to the maximum of d independent normal random variables. This allows us to conduct an hypothesis testing for extremal independence by means of the asymptotic distribution of the largest eigenvalue. Simulation studies are performed to further illustrate this approach.

This thesis focuses on stochastic processes and some of their properties are investigated which are necessary to determine the tools, the extremal index and the extremogram. Both mathematical tools measure extremal dependency within random time series. Two different models are introduced and related properties are discussed. The probability function of the Agent based model is surveyed explicitly and strong stationarity is proven. Data sets for both processes are simulated and clustering of the data is investigated with two different methods. Finally an estimation of the extremogram is used to interpret dependency of extremes within the data.

In this thesis we deal with statistical inference related to extreme value phenomena. Specifically, if X is a random vector with values in d-dimensional space, our goal is to estimate moments of ψ(X) for a suitably chosen function ψ when the magnitude of X is big. We employ the powerful tool of regular variation for random variables, random vectors and time series to formally define the limiting quantities of interests and construct the estimators. We focus on three statistical estimation problems: (i) multivariate tail estimation for regularly varying random vectors, (ii) extremogram estimation for regularly varying time series, (iii) estimation of the expected shortfall given an extreme component under a conditional extreme value model. We establish asymptotic normality of estimators for each of the estimation problems. The theoretical findings are supported by simulation studies and the estimation procedures are applied to some financial data.

In the present thesis we deal with dependence among extremal values within time series. Concerning this type of relations the commonly used autocorrelation function does not provide sufficient information. Moreover, autocorrelation function is suitable for Gaussian processes while nowadays we often work with heavy-tailed time series. In this thesis we cover two measures of extremal dependence that are used for this type of data. We introduce the coefficient of tail dependence, measure of extremal dependence based on tail characteristics of joint survival function. The second measure is called extremogram, which depends only on the extreme values in the sequence. In addition to the theoretical part, simulation study and application to real data of both described measures including their comparison are performed. Results are stated together with tables and graphical output.

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.