A characterization of D-norms and their generators based on the family of spectral functions

TL;DR

This paper characterizes D-norms and their generators in functional extreme value theory, revealing how to identify generator processes that produce specific D-norms and decomposing generators into deterministic and random parts.

Contribution

It provides a characterization of D-norms and their generators, including a decomposition into deterministic and stochastic components, advancing understanding in functional EVT.

Findings

Identifies conditions for generator processes to produce given D-norms

Shows that generators can be decomposed into deterministic and stochastic parts

Enhances methods for constructing generalized Pareto processes

Abstract

Aulbach et al. (2012) introduced the concept of D-norms in the framework of functional extreme value theory (EVT) extending the multivariate case in a natural manner. In particular, the distribution of a standard max-stable process (MSP) {\eta} \in C[0,1] is completely determined by its functional distribution function, which itself is given by some D-norm. In order to generate a generalized Pareto process (GPP) that is in the functional domain of attraction of {\eta}, one may use the fact that every D-norm is defined by some generator process with continuous sample paths. It is, however, still unknown which generator must be chosen such that a given D-norm arises. This is the content of the present paper. We will, moreover, show that a generator process may be decomposed into a functional deterministic part and a univariate random one.