Stationary max-stable fields associated to negative definite functions

TL;DR

This paper characterizes stationary max-stable processes derived from Gaussian processes with stationary increments, showing their connection to fractional Brownian motion and providing conditions for a mixed moving maxima representation.

Contribution

It establishes the law of certain max-stable processes associated with negative definite functions and characterizes their stationarity and representation as limits of Gaussian maxima.

Findings

The process $ ext{eta}$ is stationary max-stable with Gumbel margins.

The process $ ext{eta}$ arises as a limit of maxima of Gaussian processes if and only if $W$ is fractional Brownian motion.

Under certain conditions, $ ext{eta}$ admits a mixed moving maxima representation.

Abstract

Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance $\sigma^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}\delta_{U_i}$ be a Poisson point process on the real line with intensity $e^{-y} dy$. We show that the law of the random family of functions $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$, is translation invariant. In particular, the process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable process with standard Gumbel margins. The process $\eta$ arises as a limit of a suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary Gaussian processes as $n\to\infty$ if and only if $W$ is a (nonisotropic) fractional Brownian motion on ${\mathbb{R}}^d$. Under suitable conditions on $W$, the process $\eta$ has a mixed moving maxima representation.