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

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
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