Generalized Pickands constants and stationary max-stable processes

TL;DR

This paper extends the concept of Pickands constants to a broader class of stochastic processes, providing new representations and linking them to spatial extreme value theory and max-stable processes.

Contribution

It generalizes Pickands constants and their Dieker-Yakir representations to Gaussian and Lévy processes, connecting them to max-stable processes in spatial extremes.

Findings

Extended Pickands constants to Gaussian and Lévy processes

Provided new representations for these constants

Linked constants to max-stable processes in spatial extremes

Abstract

Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and L\'evy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.