## Electronic Journal of Probability

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

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 23 May 2016

https://projecteuclid.org/euclid.ejp/1464037582

Digital Object Identifier
doi:10.1214/EJP.v8-137

Mathematical Reviews number (MathSciNet)
MR1986841

Zentralblatt MATH identifier
1065.60020

#### Citation

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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