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2015 The diameter of an elliptical cloud
Yann Demichel, Ana-Karina Fermin, Philippe Soulier
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Electron. J. Probab. 20: 1-32 (2015). DOI: 10.1214/EJP.v20-3777

Abstract

We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. d-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max $U$-statistic, hence the maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obtain results in other norms for spherical vectors and we give several bi-dimensional generalizations. The main idea behind our results and their proofs is a specific property of random vectors whose norm is in the domain of attraction of the Gumbel distribution: the localization into subspaces of low dimension of vectors with a large norm.

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Yann Demichel. Ana-Karina Fermin. Philippe Soulier. "The diameter of an elliptical cloud." Electron. J. Probab. 20 1 - 32, 2015. https://doi.org/10.1214/EJP.v20-3777

Information

Accepted: 12 March 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1327.60036
MathSciNet: MR3325097
Digital Object Identifier: 10.1214/EJP.v20-3777

Subjects:
Primary: 60D05
Secondary: 60F05

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