## 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
Full Text: