Limit theorems for generalized perimeters of random inscribed polygons. II
DOI:
https://doi.org/10.21638/spbu01.2021.109Abstract
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
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References
Литература
1. Симарова Е.Н. Предельные теоремы для обобщенных периметров случайных вписанных многоугольников. I. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 7 (65), вып. 4, 678–687 (2020). https://doi.org/10.21638/spbu01.2020.409
2. Koroleva E.V., Nikitin Ya.Yu. U-max-statistics and limit theorems for perimeters and areas of random polygons. J. Multivariate Anal. 127, 99–111 (2014).
3. Прасолов В.В. Задачи по планиметрии. Изд-во МЦНМО (2006).
4. Hille E. Some geometric extremal problems. Journal of the Australian Mathematical Society 6, iss. 1, 122–128 (1966).
References
1. Simarova E.N. Limit theorems for generalized perimeters of random inscribed polygons. I. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 7 (65), iss. 4, 678–687 (2020). https://doi.org/10.21638/spbu01.2020.409 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 53, iss. 4, 434–442 (2020)].
2. Koroleva E.V., Nikitin Ya.Yu. U-max-statistics and limit theorems for perimeters and areas of random polygons. J. Multivariate Anal. 127, 99–111 (2014).
3. Prasolov V.V. Problems in plane geometry. Moscow Center for Cont. Math. Education Publ. (2006). (In Russian)
4. Hille E. Some geometric extremal problems. Journal of the Australian Mathematical Society 6, iss. 1, 122–128 (1966).
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Published
2021-05-29
How to Cite
Simarova, E. N. (2021). Limit theorems for generalized perimeters of random inscribed polygons. II. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(1), 101–110. https://doi.org/10.21638/spbu01.2021.109
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
