Generalized limit theorems for U-max statistics

Journal of Applied Probability. 2022. Vol. 59. No. 3. P. 825–848.

Yakov Nikitin, Simarova E.

U-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-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 distribution, Random area, U-max statistics, Random perimeter, Poisson approximation