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

Abstract

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 $\{U_i, i\in\mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\rm{d} u$ 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.