Limiting distributions for the maximum of a symmetric function on a random point set
type: Item
instance of: scholarly article
title: Limiting distributions for the maximum of a symmetric function on a random point set
zbMATH Open document ID: 1111.62018
published in: Journal of Theoretical Probability
publication date: 2007-02-14T00:00:00Z
review text: Let \(U_1, U_2,\ldots\) be a sequence of independent random points taking values in a measurable space \((S,\Sigma)\) according to a common probability \(P\) and let \(h:S\times S\to {\mathbb S}\) be a symmetric, Borel\(/\Sigma\times\Sigma\)-measurable function. Let \[ H_n=\max\{h(U_i, U_j): 1\leq i< j\leq n\} \] denote the maximum \(h\)-value over pairs of distinct points from \(U_1, U_2,\ldots,U_n\). Assumptions on the distribution of \(h(U_1,\cdot)\) are provided under which a continuous function of \(H_n\) converges in law to an extreme-value distribution upon suitable rescaling. The work is complementary to that appearing in \textit{M. J. B. Appel, M. J. Klass} and \textit{R. P. Russo} [J. Theor. Probab. 12, No. 1, 27--47 (1999; Zbl 0926.60018)] on the almost-sure limiting behavior of \(H_n\). In the first of two examples, the main result applied to the case of i.i.d. points distributed uniformly on the surface of a unit hypersphere in \({\mathbb R}^d\) provides the limiting distribution of the maximum pairwise distance (chord length) among the first \(n\) of the points. The second example exhibits the limiting distribution of the minimum pairwise distance among the first \(n\) of i.i.d. uniform points in a compact subset of \({\mathbb R}^d\).
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 62G32
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 5126769
zbMATH Keywords: limit distribution
zbMATH Keywords: Weibull distribution
zbMATH Keywords: random point set
zbMATH Keywords: maximum
zbMATH Keywords: extreme value
author: Martin J. B. Appel
author: Ralph P. Russo
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
full work available at URL: s10959-006-0013-6
OpenAlex ID: W1966482984
cites work: A series criterion for the almost-sure growth rate of the generalized diameter of an increasing sequence of random points
cites work: Limit laws for the diameter of a random point set
cites work: Statistical Analysis of Spherical Data
cites work: The limit distribution of the largest interpoint distance from a symmetric Kotz sample
cites work: Q5656211
cites work: Asymptotic distribution of the normal sample range
DOI: 10.1007/S10959-006-0013-6
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description: scientific article; zbMATH DE number 5126769