U-max-Statistics and Limit Theorems for Perimeters and Areas of Random Polygons

Abstract

Recently W. Lao and M. Mayer [6], [7], [9] considered $U$-max - statistics, where instead of sum appears the maximum over the same set of indices. Such statistics often appear in stochastic geometry. The examples are given by the largest distance between random points in a ball, the maximal diameter of a random polygon, the largest scalar product within a sample of points, etc. Their limit distribution is related to the distribution of extreme values. Among the interesting results obtained in [6], [7], [9] are limit theorems for the maximal perimeter and the maximal area of random triangles inscribed in a circumference. In the present paper we generalize these theorems to convex $m$-polygons, $m \geq 3,$ with random vertices on the circumference. Next, a similar problem is solved for the minimal perimeter and the minimal area of circumscribed $m$-polygons which has not been previously considered in literature. Finally, we discuss the obtained results when $m \to \infty.$