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

Consider the max-stable process $\eta(t) = \max_{i\in\mathbb N} U_i \rm{e}^{\langle X_i, t\rangle - \kappa(t)}$,$t\in$\mathbb{R}$^d$,where${U_i, i\in$\mathbb{N}$}$arepointsofthePoissonprocesswithintensity$u^{-2}\rm{d} u$on$(0,\infty)$,$X_i$,$i\in$\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 $\eta$ 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.