×

Limit theorems for the number of maxima in random samples from planar regions. (English) Zbl 0986.60007

Consider a set of \(n\) points in the plane. A point dominates another point if both coordinates are at least as large as those of the second point. The maxima of the \(n\) points are those points which are not dominated by any other point in the set. For a given convex polygon let the \(n\) points be independently and uniformly distributed in the polygon. Let \(M\) be the number of maximal points. A central limit theorem is proved for \(M\) as \(n\) tends to infinity. The proof is done by reducing the general case to a triangle with corners \((0,0)\), \((1,0)\), \((0,1)\), and then using the method of moments. Also planar regions bounded above by nondecreasing functions are considered and Poisson approximation results are obtained. For the derivations explicit rather intricate calculations are used. Many references are provided.

MSC:

60D05 Geometric probability and stochastic geometry
60C05 Combinatorial probability
PDF BibTeX XML Cite
Full Text: DOI EuDML EMIS