Stationary max-stable fields associated to negative definite functions

Abstract

Let Wi, i ∈ N{double struck}, be independent copies of a zero-mean Gaussian process {W(t), t ∈ R{double struck}d} with stationary increments and variance σ2(t). Independently of Wi, let ∑∞i=1 δUi 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 {Vi(·), i ∈ N{double struck}}, where Vi(t) = Ui + Wi(t) - σ2(t)/2, is translation invariant. In particular, the process η(t) = V∞i=1 Vi(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 R{double struck}d. Under suitable conditions on W, the process η has a mixed moving maxima representation. © Institute of Mathematical Statistics, 2009.