Equivalent Representations of Max-Stable Processes via $\ell^p$ Norms

TL;DR

This paper introduces a family of $ ext{ell}^p$ norm-based representations for max-stable processes, unifying existing models and providing conditions for their existence, along with properties like ergodicity and mixing.

Contribution

It presents a new unified framework for representing max-stable processes using $ ext{ell}^p$ norms, including formulae for switching representations and conditions for their existence.

Findings

Unified $ ext{ell}^p$ norm-based representations for max-stable processes.

Derived conditions for the existence of $ ext{ell}^p$ representations.

Analyzed properties such as ergodicity and mixing of the processes.

Abstract

While max-stable processes are typically written as pointwise maxima over an infinite number of stochastic processes, in this paper, we consider a family of representations based on $\ell^p$ norms. This family includes both the construction of the Reich-Shaby model and the classical spectral representation by de Haan as special cases. As the representation of a max-stable process is not unique, we present formulae to switch between different equivalent representations. We further provide a necessary and sufficient condition for the existence of a $\ell^p$ norm based representation in terms of the stable tail dependence function of a max-stable process. Finally, we discuss several properties of the represented processes such as ergodicity or mixing.