Stationary max-stable fields associated to negative definite functions

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.

Published in at http://dx.doi.org/10.1214/09-AOP455 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• On spatial extremes: with application to a rainfall problem