Stationary max-stable fields associated to negative definite functions

Abstract

Let W_i, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝ^d} with stationary increments and variance σ²(t). Independently of W_i, let ∑_i=1^∞δ_Ui 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(⋅), i∈ℕ}, where V_i(t)=U_i+W_i(t)−σ²(t)/2, is translation invariant. In particular, the process η(t)=⋁_i=1^∞V_i(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^d. Under suitable conditions on W, the process η has a mixed moving maxima representation.