This article is cited in 2 papers

MATHEMATICS

Limit theorems for generalized perimeters of random inscribed polygons. II

E. N. Simarova<sup>ab

a</sup> St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
^b Leonhard Euler International Mathematical Institute, 29B, 14 liniya V. O., St. Petersburg, 199178, Russian Federation

Abstract: 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.

Keywords: U-max statistics, limit behavior, uniform distribution on a circle, the sum of the degrees of the sides of the polygon.

UDC: 519.224

MSC: 60D05, 60F05, 60G70

Received: 01.03.2020
Revised: 28.06.2020
Accepted: 17.09.2020

DOI: 10.21638/spbu01.2021.109

 English version: Vestnik St. Petersburg University, Mathematics, 2021, 8:3, 78–85