U-max-Statistics and limit Theorems for perimeters and areas of random polygons
Recently Lao and Mayer (2008) considered U-max-statistics, where the maximum of kernels over the set of indices is studied instead of the usual sums. Such statistics emerge frequently in stochastic geometry. The examples include 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 distributions are related to the distributions of extreme values. Among the results obtained by Lao and Mayer, the limit theorems for the maximal perimeter and the maximal area of random triangles inscribed in a circumference are of great interest. In the present paper, we generalize these theorems to the case of convex m-polygons, m ≥ 3, with random vertices on the circumference. In addition, a similar problem for the minimal perimeter and the minimal area of circumscribed m-polygons is solved in this paper. This problem has not been studied in the literature so far.