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September 2009 Stationary max-stable fields associated to negative definite functions
Zakhar Kabluchko, Martin Schlather, Laurens de Haan
Ann. Probab. 37(5): 2042-2065 (September 2009). DOI: 10.1214/09-AOP455


Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1δUi be a Poisson point process on the real line with intensity eydy. We show that the law of the random family of functions {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=i=1Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.


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Zakhar Kabluchko. Martin Schlather. Laurens de Haan. "Stationary max-stable fields associated to negative definite functions." Ann. Probab. 37 (5) 2042 - 2065, September 2009.


Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1208.60051
MathSciNet: MR2561440
Digital Object Identifier: 10.1214/09-AOP455

Primary: 60G70
Secondary: 60G15

Keywords: Extremes , Gaussian processes , Poisson point processes , Stationary max-stable processes

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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