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Generalized limit theorems for U-max statistics

Part of: Limit theorems Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  11 July 2022

Yakov Nikitin  and
Ekaterina Simarova Open the ORCID record for Ekaterina Simarova [Opens in a new window]
Yakov Nikitin*
Affiliation:
Saint-Petersburg State University
Ekaterina Simarova*
Affiliation:
Saint-Petersburg State University and Leonhard Euler International Mathematical Institute
*
*Postal address: Department of Mathematics and Mechanics, Universitetsky pr. 28, Stary Peterhof 198504, Russia.
*Postal address: Department of Mathematics and Mechanics, Universitetsky pr. 28, Stary Peterhof 198504, Russia.
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Abstract

$U{\hbox{-}}\textrm{max}$ statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for $U{\hbox{-}}\textrm{max}$ statistics is via a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum, and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.

Keywords

Weibull distributionPoisson approximationU-max statisticsrandom perimeterrandom area

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 825 - 848
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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