This article is cited in 2 papers

MATHEMATICS

Limit theorems for generalized perimeters of random inscribed polygons. I

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 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.

Keywords: U-max-statistics, Poisson approximation, generalized perimeter, limiting behavior.

UDC: 519.224

MSC: 60D05, 60F05, 60G70

Received: 05.03.2020
Revised: 18.05.2020
Accepted: 18.07.2020

DOI: 10.21638/spbu01.2020.409

 English version: Vestnik St. Petersburg University, Mathematics, 2020, 7:4, 434–442