Electronic Journal of Probability

Berry-Esseen Bounds for the Number of Maxima in Planar Regions

Zhi-Dong Bai, Hsien-Kuei Hwang, and Tsung-Hsi Tsai

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We derive the optimal convergence rate $O(n^{-1/4})$ in the central limit theorem for the number of maxima in random samples chosen uniformly at random from the right equilateral triangle with two sides parallel to the axes, the hypotenuse with the slope $-1$ and consituting the top part of the boundary of the triangle. A local limit theorem with rate is also derived. The result is then applied to the number of maxima in general planar regions (upper-bounded by some smooth decreasing curves) for which a near-optimal convergence rate to the normal distribution is established.

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Electron. J. Probab., Volume 8 (2003), paper no. 9, 26 p.

First available in Project Euclid: 23 May 2016

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Dominance Maximal points Central limit theorem Berry-Esseen bound Local limit theorem Method of moments


Bai, Zhi-Dong; Hwang, Hsien-Kuei; Tsai, Tsung-Hsi. Berry-Esseen Bounds for the Number of Maxima in Planar Regions. Electron. J. Probab. 8 (2003), paper no. 9, 26 p. doi:10.1214/EJP.v8-137. https://projecteuclid.org/euclid.ejp/1464037582

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  • Z.-D. Bai, C.-C. Chao, H.-K. Hwang and W.-Q. Liang (1998). On the variance of the number of maxima in random vectors and its applications, Annals of Applied Probability, 8, 886-895.
  • Z.-D. Bai, H.-K. Hwang, W.-Q. Liang, and T.-H. Tsai (2001). Limit theorems for the number of maxima in random samples from planar regions, Electronic Journal of Probability, 6, paper no. 3, 41 pages.
  • Yu. V. Prohorov (1953), Asymptotic behavior of the binomial distribution, in Selected Translations in Mathematical Statistics and Probability, Vol. 1, pp. 87-95, ISM and AMS, Providence, R.I. (1961); translation from Russian of: Uspehi Matematiceskih Nauk, 8 (1953), no. 3 (35), 135-142.
  • A. D. Barbour and A. Xia (2001). The number of two dimensional maxima, Advances in Applied Probability, 33, 727-750.
  • Y. Baryshnikov (2000). Supporting-points processes and some of their applications, Probability Theory and Related Fields, 117, 163-182.
  • R. A. Becker, L. Denby, R. McGill and A. R. Wilks (1987). Analysis of data from the "Places Rated Almanac", The American Statistician, 41, 169-186.
  • A. J. Cabo and P. Groeneboom (1994). Limit theorems for functionals of convex hulls, Probability Theory and Related Fields, 100, 31-55.
  • T. M. Chan (1996). Output-sensitive results on convex hulls, extreme points, and related problems, Discrete and Computational Geometry, 16, 369-387.
  • S. N. Chiu and M. P. Quine, Central limit theory for the number of seeds in a growth model in Rd with inhomogeneous Poisson arrivals, Annals of Applied Probability, 7 (1997), 802-814.
  • A. Datta and S. Soundaralakshmi (2000), An effcient algorithm for computing the maximum empty rectangle in three dimensions, Information Sciences, 128, 43-65.
  • L. Devroye (1993). Records, the maximal layer, and the uniform distributions in monotone sets, Computers and Mathematics with Applications, 25, 19-31.
  • M. E. Dyer and J. Walker (1998). Dominance in multi-dimensional multiple-choice knapsack problems, Asia-Pacific Journal of Operational Research, 15, 159-168.
  • I. Z. Emiris, J. F. Canny and R. Seidel (1997). Effcient perturbations for handling geometric degeneracies, Algorithmica, 19, 219-242.
  • J. L. Ganley (1999). Computing optimal rectilinear Steiner trees: A survey and experimental evaluation, Discrete Applied Mathematics, 90, 161-171.
  • M. J. Golin (1993). Maxima in convex regions, in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (Austin, TX, 1993), 352-360, ACM, New York.
  • P. Groeneboom (1988), Limit theorems for convex hulls, Probability Theory and Related Fields, 79, 327-368.
  • H.-K. Hwang (2002). Second phase changes in random $m$-ary search trees and generalized quicksort: convergence rates, Annals of Probability, 31, 609-629.
  • R. E. Johnston and L. R. Khan (1995). A note on dominance in unbounded knapsack problems, Asia-Paciffic Journal of Operational Research, 12, 145-160.
  • S. Martello and P. Toth (1990). Knapsack Problems: Algorithms and Computer Implementations, John Wiley & Sons, New York.
  • R. Neininger and L. Rüschendorf (2002). A general contraction theorem and asymptotic normality in combinatorial structures, Annals of Applied Probability, accepted for publication (2003).
  • V. V. Petrov (1975). Sums of Independent Random Variables, Springer-Verlag, New York.
  • M. Zachariasen (1999). Rectilinear full Steiner tree generation, Networks, 33, 125-143.