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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

Information

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.

References

Aldous, D. (1978). Stopping times and tightness. Ann. Prob. 6, 335–340.CrossRefGoogle Scholar
Bai, Z.-D., Chao, C.-C., Hwang, H.-K. and Liang, W.-Q. (1998). On the variance of the number of maxima in random vectors and its applications. Ann. Appl. Prob. 8, 886–895.CrossRefGoogle Scholar
Bai, Z.-D., Hwang, H.-K., Liang, W.-Q. and Tsai, T.-H. (2001). Limit theorems for the number of maxima in random samples from planar regions. Tech. Rep. C-2000-05, Institute of Statistical Science, Academia Sinica, Taiwan. Electron. J. Prob. 6, No. 3. Available at http://www.math.washington.edu/simejpecp/.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
Barbour, A. D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combinatorial Theory B 47, 125–145.CrossRefGoogle Scholar
Baryshnikov, Y. (2000). Supporting-points processes and some of their applications. Prob. Theory Relat. Fields 117, 163–182.CrossRefGoogle Scholar
Chen, L. H. Y. (1986). The rate of convergence in a central limit theorem for dependent random variables with arbitrary index set. Tech. Rep., IMA, Minnesota.Google Scholar
Chiu, S. N. and Lee, H. Y. (2000). A regularity condition and strong limit theorems for linear birth–growth systems. To appear in Math. Nachr.Google Scholar
Chiu, S. N. and Quine, M. P. (1997). Central limit theory for the number of seeds in a growth model in Rd with inhomogeneous Poisson arrivals. Ann. Appl. Prob. 7, 802–814.CrossRefGoogle Scholar
Devroye, L. (1993). Records, the maximal layer, and uniform distributions in monotone sets. Comput. Math. Applic. 25, 19–31.CrossRefGoogle Scholar
Holst, L., Quine, M. P. and Robinson, J. (1996). A general stochastic model for nucleation and linear growth. Ann. Appl. Prob. 6, 903–921.CrossRefGoogle Scholar
Ivanoff, G. and Merzbach, E. (2000). Set-indexed martingales. Chapman and Hall, Boca Raton, FL.Google Scholar
Quine, M. P. and Robinson, J. (1990). A linear random growth model. J. Appl. Prob. 27, 499–509.CrossRefGoogle Scholar
Rényi, A., (1962). On the extreme elements of observations. Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 12, 105–121 (in Hungarian). English translation: in Selected Papers of Alfréd Rényi, Vol. 3, eds Turán, P., Akadémiai Kiadó, Budapest, 1976, pp. 50–66.Google Scholar