Limit theorems for generalized perimeters of random inscribed polygons. II

Lao and Mayer (2008) recently developed the theory of U-max statistics, where instead of the usual sums over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Examples include the greatest distance between random points in a ball, the maximum diameter of a random polygon, the largest scalar product in a sample of points, etc. Their limit distributions are related to distribution of extreme values. This is the second article devoted to the study of the generalized perimeter of a polygon and the limit behavior of the U-max statistics associated with the generalized perimeter. Here we consider the case when the parameter y, arising in the definition of the generalized perimeter, is greater than 1. The problems that arise in the applied method in this case are described. The results of theorems on limit behavior in the case of a triangle are refined.

Limit theorems for generalized perimeters of random inscribed polygons. I

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.