Soutenance publique de thèse

Pour l’obtention du grade de Docteur en Sciences

par Monsieur Thomas MEINGUET

Licencié en sciences mathématiques

"Heavy tailed functional time series"

Mardi 31 août 2010 à 10h00
Salle du Sénat académique
Halles Universitaires
Place de l’Université, 1
1348 Louvain-la-Neuve

Membres du jury :

Professeur J. SEGERS (UCL) (promoteur)

Professeur M. DENUIT (UCL) (promoteur)

Professeur R. VON SACHS (UCL) (président)

Professeur I. VAN KEILEGOM (UCL)

Professeur J. BEIRLANT (KUL)

Professeur H. HULT (KTH – Suède)

Abstract:

The goal of this thesis is to treat the temporal tail dependence and the crosssectional tail dependence of heavy tailed functional time series. Functional time series are aimed at modelling spatio-temporal phenomena; for instance rain, temperature, pollution on a given geographical area, with temporally dependent observations. Heavy tails mean that the series can exhibit much higher spikes than with Gaussian distributions for instance. In such cases, second moments cannot be assumed to exist, violating the basic assumption in standard functional data analysis based on the sequence of autocovariance operators.
While originally defined for deterministic univariate functions and random variables, the concept of regular variation has by now been defined and studied in quite abstract settings. As for random variables, regular variation provides the mathematical backbone for a coherent theory of extreme values. By considering our functional observations as points in a suitable function space, we are led to consider regularly varying time series taking values in Banach spaces.
The main tools introduced in this thesis for a regularly varying functional time series are its tail process and its spectral process. These objects capture all the aspects of the probability distribution of extreme values jointly over time and space. In fact, their existence is equivalent to the regular variation of the series.
The development of the tail and spectral process for heavy tailed functional time series is followed by three theoretical applications.
The first application is a characterization of a variety of indices and objects describing the extremal behavior of the series: the extremal index, tail dependence coefficients, the extremogram and the point process of extremes.
The second is the computation of an explicit expression of the tail and spectral processes for heavy tailed linear functional time series.
The third and final application is the introduction and the study of a model for the spatio-temporal dependence for functional time series called maxima of moving maxima of continuous functions (CM3 processes), with the development of an estimation method.