Representations of Max-Stable Processes via Exponential Tilting

TL;DR

This paper introduces new representations of max-stable processes using exponential tilting, extending existing models, and provides applications including stationarity conditions and characterizations of Gaussian processes.

Contribution

It develops exponential tilting representations for max-stable processes, extending Dieker & Mikosch's work, and offers new formulas and characterizations related to Gaussian and max-stable processes.

Findings

New representations of max-stable processes via exponential tilting.

Conditions for stationarity of max-stable processes derived.

Alternative proof of Gaussian processes with stationary increments.

Abstract

The recent contribution Dieker & Mikosch (2015) [1] obtained important representations of max-stable stationary Brown-Resnick random fields $\zeta_Z$ with a spectral representation determined by a Gaussian process $Z$. With motivations from \cite{DM} we derive for some general $Z$, representations for $\zeta_Z$ via exponential tilting of $Z$. Our main findings concern a) Dieker-Mikosch representations of max-stable processes, b) two-sided extensions of stationary max-stable processes, c) inf-argmax representation of any max-stable distribution, and d) new formulas for generalised Pickands constants. Our applications include new conditions for the stationarity of $\zeta_Z$, a characterisation of Gaussian random vectors and an alternative proof of Kabluchko's characterisation of Gaussian processes with stationary increments.