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The number of two-dimensional maxima

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

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
A. Xia
A. D. Barbour*
Affiliation:
Universität Zürich
A. Xia*
Affiliation:
University of Melbourne
*
∗ Postal address: Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: adb@amath.unizh.ch
∗∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
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Abstract

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.

Keywords

Maximal pointsStein's methodrecord valuesJohnson-Mehl process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 727 - 750
Copyright
Copyright © Applied Probability Trust 2001 

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Footnotes

Partly supported by Schweizerischer Nationalfondsprojekte 20-50686.97 and 20-61753.00.

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