Characteristic functions as offshoots of multivariate extreme value theory

Multivariate extreme value theory, namely the definition and study of extreme events in several dimensions, has been largely developed over the last 30 years. One particularly interesting framework for multivariate extreme value theory is obtained by extending the univariate definition of extreme value distributions, found from the convergence of linearly normalised sample maxima, to the multivariate case via the use of the componentwise maximum operator. This gives rise to the class of multivariate max-stable distribution functions. A theory of multivariate regular variation, at the heart of which the angular measure lies, can then be written and constitutes a reasonable framework for inference about extreme events in multidimensional settings. 

Although this is by no means obvious, a common thread through multivariate extreme value theory can be found using the notion of D-norm: loosely, there is a simple, explicit one-to-one correspondence between max-stable distributions with negative exponential margins and the set of D-norms. These norms have mainly been studied for their potential to unify the presentation of multivariate extreme value theory and trivialise certain proofs, such as those of Takahashi's characterisations of complete dependence or independence of the components of a max-stable distribution. As a mathematical object themselves, however, D-norms are interesting as well, and their theory remains in full development. I will here introduce the notion of D-norm and then present two offshoots of multivariate extreme value theory inspired by this concept, which I will call the max- and min-characteristic function. I will present some interesting properties of such functions (they characterise distributions, are compatible with weak convergence, have simple closed forms for Generalised Pareto distributions...) and will discuss potential applications in statistics. [Joint work with Michael Falk, University of Wuerzburg]