The Annals of Applied Probability

Asymptotic distribution of the maximum interpoint distance in a sample of random vectors with a spherically symmetric distribution

Sreenivasa Rao Jammalamadaka and Svante Janson

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Abstract

Extreme value theory is part and parcel of any study of order statistics in one dimension. Our aim here is to consider such large sample theory for the maximum distance to the origin, and the related maximum “interpoint distance,” in multidimensions. We show that for a family of spherically symmetric distributions, these statistics have a Gumbel-type limit, generalizing several existing results. We also discuss the other two types of limit laws and suggest some open problems. This work complements our earlier study on the minimum interpoint distance.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3571-3591.

Dates
Received: December 2012
Revised: November 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703782

Digital Object Identifier
doi:10.1214/14-AAP1082

Mathematical Reviews number (MathSciNet)
MR3404644

Zentralblatt MATH identifier
1328.60027

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory

Keywords
Maximum interpoint distance extreme value distributions Gumbel distribution

Citation

Jammalamadaka, Sreenivasa Rao; Janson, Svante. Asymptotic distribution of the maximum interpoint distance in a sample of random vectors with a spherically symmetric distribution. Ann. Appl. Probab. 25 (2015), no. 6, 3571--3591. doi:10.1214/14-AAP1082. https://projecteuclid.org/euclid.aoap/1443703782


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