A characterization of the normal distribution using stationary max-stable processes

Consider the max-stable process $\eta(t) = \max_{i\in\mathbb N} U_i \rm{e}^{\langle X_i, t\rangle - \kappa(t)}$, $t\in\mathbb{R}^d$, where $\{U_i, i\in\mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\rm{d} u$ on $(0,\infty)$, $X_i$, $i\in\mathbb{N}$, are independent copies of a random $d$-variate vector $X$ (that are independent of the Poisson process), and $\kappa: \mathbb{R}^d \to \mathbb{R}$ is a function. We show that the process $\eta$ is stationary if and only if $X$ has multivariate normal distribution and $\kappa(t)-\kappa(0)$ is the cumulant generating function of $X$. In this case, $\eta$ is a max-stable process introduced by R. L. Smith

Brown-Resnick Processes: Analysis, Inference and Generalizations

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