## Electronic Journal of Probability

### The diameter of an elliptical cloud

#### 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

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

Rights

#### 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

#### References

• Balkema, Guus; Embrechts, Paul. High risk scenarios and extremes. A geometric approach. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2007. xiv+375 pp. ISBN: 978-3-03719-035-7.
• Philippe Barbe and Miriam Seifert. A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values. arXiv:1311.0540, 2013.
• Fougères, Anne-Laure; Soulier, Philippe. Limit conditional distributions for bivariate vectors with polar representation. Stoch. Models 26 (2010), no. 1, 54–77.
• Korshunov, D. A.; Piterbarg, V. I.; Khashorva, E. On the extreme values of Gaussian chaos. (Russian) Dokl. Akad. Nauk 452 (2013), no. 5, 483–485; translation in Dokl. Math. 88 (2013), no. 2, 566–568
• Hsing, Tailen; Rootzén, Holger. Extremes on trees. Ann. Probab. 33 (2005), no. 1, 413–444.
• Jammalamadaka, S. Rao; Janson, Svante. Limit theorems for a triangular scheme of $U$-statistics with applications to inter-point distances. Ann. Probab. 14 (1986), no. 4, 1347–1358.
• S. Rao Jammalamadaka and Svante Janson. Asymptotic distribution of the maximum interpoint distance in a sample of random vectors with a spherically symmetric distribution. arXiv:1211.0822, 2012.
• Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2.
• Resnick, Sidney I. Extreme values, regular variation, and point processes. Applied Probability. A Series of the Applied Probability Trust, 4. Springer-Verlag, New York, 1987. xii+320 pp. ISBN: 0-387-96481-9.