STATIONARY MAX-STABLE FIELDS ASSOCIATED TO NEGATIVE DEFINITE FUNCTIONS

Let W-i, i is an element of N, be independent copies of a zero-mean Gaussian process {W (t), t is an element of R-d} with stationary increments and variance sigma(2)(t). Independently of W-i, let Sigma(infinity)(i=1) U-delta(i) 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 is an element of N}, where V-i(t) = U-i + W-i(t) - sigma(2)(t)/2, is translation invariant. In particular, the process n(t) = V-i=1(infinity) V-i(t) is a stationary max-stable process with standard Gumbel margins. The process eta arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n -> infinity if and only if W is a (nonisotropic) fractional Brownian motion on R-d. Under suitable conditions on W, the process eta has a mixed moving maxima representation.