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

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.

17 pages

References in corpus (1)

• A hierarchical max-stable spatial model for extreme precipitation