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

2015/08/18 by Engelke, Sebastian, Kabluchko, Zakhar
#60G15 #60G70 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1508.04266

Abstract

Consider the max-stable process η(t) = max_{i∈\mathbb N} U_i \rme^{⟨ X_i, t⟩ - κ(t)}, t∈ℝ^d, where \U_i, i∈ℕ\ are points of the Poisson process with intensity u^-2\rmd u on (0,∞), X_i, i∈ℕ, are independent copies of a random d-variate vector X (that are independent of the Poisson process), and κ: ℝ^d → 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.

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