Berry-Esseen Bounds for the Number of Maxima in Planar Regions
Zhi-Dong Bai, National University of Singapore and Northeast Normal University
Hsien-Kuei Hwang, Academia Sinica, Taipei
Tsung-Hsi Tsai, Academia Sinica, Taipei
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 triangle of the shape
. 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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