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

Abstract

Consider a max-stable process of the form $\eta (t) = \max _{i\in \mathbb {N}} U_{i} \mathrm {e}^{\langle X_{i}, t\rangle - \kappa (t)}$, $t\in \mathbb {R}^{d}$, where $\{U_{i}, i\in \mathbb {N}\}$ are points of the Poisson process with intensity u ⁻²du on (0,∞), X _i , $i\in \mathbb {N}$, are independent copies of a random d-variate vector X (that are independent of the Poisson process), and $\kappa :\mathbb {R}^{d} \to \mathbb {R}$ is a function. We show that the process η is stationary if and only if X has multivariate normal distribution and κ(t)−κ(0) is the cumulant generating function of X. In this case, η is a max-stable process introduced by R. L. Smith.