Electronic Journal of Probability

The diameter of an elliptical cloud

Yann Demichel, Ana-Karina Fermin, and Philippe Soulier

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 27, 32 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067133

Digital Object Identifier
doi:10.1214/EJP.v20-3777

Mathematical Reviews number (MathSciNet)
MR3325097

Zentralblatt MATH identifier
1327.60036

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F05: Central limit and other weak theorems

Keywords
Elliptical Distributions Interpoint Distance Extreme Value Theory Gumbel Distribution

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Demichel, Yann; Fermin, Ana-Karina; Soulier, Philippe. The diameter of an elliptical cloud. Electron. J. Probab. 20 (2015), paper no. 27, 32 pp. doi:10.1214/EJP.v20-3777. https://projecteuclid.org/euclid.ejp/1465067133


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