A characterization of the normal distribution using stationary max-stable processes

Consider a max-stable process of the form , , where are points of the Poisson process with intensity u (-2)du on (0,a), X (i) , , are independent copies of a random d-variate vector X (that are independent of the Poisson process), and is a function. We show that the process eta is stationary if and only if X has multivariate normal distribution and kappa(t)-kappa(0) is the cumulant generating function of X. In this case, eta is a max-stable process introduced by R. L. Smith.