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The limit distribution of the largest interpoint distance for distributions supported by ad-dimensional ellipsoid and generalizations

Part of: Distribution theory Stochastic processes Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  11 January 2017

Michael Schrempp
Michael Schrempp*
Affiliation:
Karlsruhe Institute of Technology
*
* Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, Englerstr. 2, D-76131 Karlsruhe, Germany. Email address: [email protected]
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Abstract

We study the asymptotic behaviour of the maximum interpoint distance of random points in a d-dimensional ellipsoid with a unique major axis. Instead of investigating only a fixed number of n points as n tends to ∞, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. Our main result covers the case of uniformly distributed points.

Keywords

Maximum interpoint distancegeometric extreme value theoryPoisson processuniform distribution in an ellipsoid

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1256 - 1270
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Appel, M. J. B., Najim, C. A. and Russo, R. P. (2002). Limit laws for the diameter of a random point set. Adv. Appl. Prob. 34, 1–10.Google Scholar
[2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York>.Google Scholar
[3] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, General Theory and Structure, 2nd edn. Springer, New York.Google Scholar
[4] Demichel, Y., Fermin, A.-K. and Soulier, P. (2015). The diameter of a random elliptical cloud. Electron. J. Prob. 20, 27, 32pp.Google Scholar
[5] Henze, N. and Klein, T. (1996). The limit distribution of the largest interpoint distance from a symmetric Kotz sample. J. Multivariate Analysis 57, 228–239.Google Scholar
[6] Henze, N. and Lao, W. (2010). The limit distribution of the largest interpoint distance for power-tailed spherically decomposable distributions and their affine images. Preprint, Karlsruhe Institute of TechnologyGoogle Scholar
[7] Jammalamadaka, S. R. and Janson, S. (2015). Asymptotic distribution of the maximum interpoint distance in a sample of random vectors with a spherically symmetric distribution. Ann. Appl. Prob. 25, 3571–3591.Google Scholar
[8] Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie, Berlin.CrossRefGoogle Scholar
[9] Lao, W. (2010). Some weak limit laws for the diameter of random point sets in bounded regions, KIT Scientific Publishing, Karlsruhe.Google Scholar
[10] Matthews, P. C. and Rukhin, A. L. (1993). Asymptotic distribution of the normal sample range. Ann. Appl. Prob. 3, 454–466.CrossRefGoogle Scholar
[11] Mayer, M. and Molchanov, I. (2007). Limit theorems for the diameter of a random sample in the unit ball. Extremes 10, 129–150.CrossRefGoogle Scholar
[12] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes, Springer, New York.Google Scholar
[13] Schrempp, M. (2015). The limit distribution of the largest interpoint distance for distributions supported by an ellipse and generalizations. Preprint. Available at http://arxiv.org/abs/1505.01597.Google Scholar