Stationary max-stable fields associated to negative definite functions

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 ?2(t). Independently of Wi, let ?? i=1 ?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 {Vi(·), i ? N{double struck}}, where Vi(t) = Ui + Wi(t) - ?2(t)/2, is translation invariant. In particular, the process ?(t) = V? i=1 Vi(t) is a stationary max-stable process with standard Gumbel margins. The process ? 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 ? has a mixed moving maxima representation.