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
