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

Abstract

Consider the max-stable process $\eta (t)={\max }_{i\in \mathbb{N}}{U}_i{\mathrm{e}}^{⟨{\mathrm{X}}_{\mathrm{i}},\mathrm{t}⟩-\kappa (\mathrm{t})}$, $t\in {\mathbb{R}}^d$, where $\{U_i,i\in \mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\mathrm{d}\mathrm{u}$ 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.