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Extremes

, Volume 19, Issue 1, pp 1–6 | Cite as

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

  • Sebastian EngelkeEmail author
  • Zakhar Kabluchko
Article

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.

Keywords

Smith max-stable process Stationarity Extreme value theory Multivariate normal distribution 

AMS 2000 Subject Classifications

60G70 60G15 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Ecole Polytechnique Fédérale de Lausanne, EPFL-FSB-MATHAA-STATLausanneSwitzerland
  2. 2.Institute of Mathematical StatisticsUniversity of MünsterMünsterGermany

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