Stationary max-stable fields associated to negative definite functions

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=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, where Vi (t) =Ui+Wi (t) −σ2 (t) /2, is translation invariant. In particular, the process η (t) =⋁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 ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.