Limit theorems for generalized perimeters of random inscribed polygons

Abstract

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.