Consider a cloud of i.i.d. random vectors in ${\Bbb R}^d$, $d\geq1$. What is the asymptotic behavior of the diameter of the cloud when the number of points in the cloud tends to infinity? This question is a topic of interest for about twenty years but remains open in many general settings. Results obtained are deeply related to the distribution of the random vectors, in particular they depend on whether the distribution is supported by a bounded set or not. In this talk, we consider a cloud of elliptical vectors whose maximum Euclidean norm is in the domain of attraction of the Gumbel distribution. Since the diameter is a maximum of random but dependent variables, this question is a non standard extreme value problem. In particular, the limiting distributions may not be extreme value distributions. Using a *localization principle* for vectors with large norm we obtain exhaustive results and precise descriptions of the limiting distributions. Finally, we use this principle again to investigate some natural generalizations.

This is a joint work with Ana-Karina Fermin and Philippe Soulier.

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