The Annals of Probability

Stationary max-stable fields associated to negative definite functions

Zakhar Kabluchko, Martin Schlather, and Laurens de Haan

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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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Ann. Probab. Volume 37, Number 5 (2009), 2042-2065.

First available in Project Euclid: 21 September 2009

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Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes

Stationary max-stable processes Gaussian processes Poisson point processes extremes


Kabluchko, Zakhar; Schlather, Martin; de Haan, Laurens. Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37 (2009), no. 5, 2042--2065. doi:10.1214/09-AOP455.

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