Berry-{E}sseen bounds for the number of maxima in planar regions. (English) Zbl 1065.60020

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

Electron. J. Probab. 8, Paper No. 9, 26 p. (2003).

Summary: 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 triangle of the shape with corners \((0,0), (0,1), (1,0)\). 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.

MSC:

60F05	Central limit and other weak theorems
60D05	Geometric probability and stochastic geometry

Keywords:

central limit theorem; maxima in random samples; near-optimal convergence rate