# Title: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.
 Comments: Published in at this http URL the Annals of Probability (this http URL) by the Institute of Mathematical Statistics (this http URL) Subjects: Probability (math.PR) MSC classes: 60G70 (Primary) 60G15 (Secondary) Journal reference: Annals of Probability 2009, Vol. 37, No. 5, 2042-2065 DOI: 10.1214/09-AOP455 Report number: IMS-AOP-AOP455 Cite as: arXiv:0806.2780 [math.PR] (or arXiv:0806.2780v3 [math.PR] for this version)

## Submission history

From: Zakhar Kabluchko [view email]
[v1] Tue, 17 Jun 2008 13:27:44 UTC (339 KB)
[v2] Mon, 27 Apr 2009 12:14:45 UTC (350 KB)
[v3] Fri, 25 Sep 2009 11:50:43 UTC (121 KB)
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