On Max-Stable Processes and the Functional D-Norm

TL;DR

This paper develops a new framework for analyzing max-stable processes using a functional D-norm, extending classical multivariate extreme value theory to functional spaces and characterizing their domains of attraction.

Contribution

It introduces a functional domain of attraction approach, defines a distribution function G via a D-norm, and extends the concept of GPDs to the functional setting, providing new characterizations.

Findings

Representation of G via D-norm in functional space

Characterization of domain of attraction for copula processes

Introduction of ta-neighborhoods with polynomial convergence rates

Abstract

We introduce a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions (GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case. Applying this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD. \delta-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known in the multivariate case.