Brown-Resnick Processes: Analysis, Inference and Generalizations

Abstract

This thesis deals with the analysis, inference and further generalizations of a rich and flexible class of max-stable random fields, the so-called Brown-Resnick processes. The first chapter gives the explicit distribution of the shape functions in the mixed moving maxima representation of the original Brown-Resnick process based on Brownian motions. The result is particularly useful for a fast simulation method. In chapter 2, a multivariate peaks-over-threshold approach for parameter estimation of Hüsler-Reiss distributions, a popular model in multivariate extreme value theory, is presented. As Hüsler-Reiss distributions constitute the finite dimensional margins of Brown-Resnick processes based on Gaussian random fields, the estimators directly enable statistical inference for this class of max-stable processes. As an application, a non-isotropic Brown-Resnick process is fitted to the extremes of 12-year data of daily wind speed measurements. Chapter 3 is concerned with the definition of Brown-Resnick processes based on Lévy processes on the whole real line. Amongst others, it is shown that these Lévy-Brown-Resnick processes naturally arise as limits of maxima of stationary stable Ornstein-Uhlenbeck processes. The last chapter is devoted to the study of maxima of d-variate Gaussian triangular arrays, where in each row the random vectors are assumed to be independent, but not necessarily identically distributed. The row-wise maxima converge to a new class of multivariate max-stable distributions, which can be seen as max-mixtures of Hüsler-Reiss distributions