Equivalent representations of max-stable processes via ℓp-norms. (28th March 2018)

• Record Type: Journal Article
• Authors: Oesting, Marco
• Is Part Of: Journal of applied probability. Volume 55:Number 1(2018)
• Page Start: 54
• Page End: 68
• Publication Date: 2018-03-28
• Subjects: Extreme value theory, -- Reich–Shaby model, -- spectral representation, -- stable tail dependence function; Primary 60G70, -- Secondary 60G52; 519.2
• DOI: 10.1017/jpr.2018.5
• Languages: English
• ISSNs: 0021-9002

Abstract

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 ℓ p -norms. This family includes both the construction of the Reich–Shaby model and the classical spectral representation by de Haan (1984) 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 an ℓ 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.