Mathematics and Statistics Vol. 11(6), pp. 960 - 964
DOI: 10.13189/ms.2023.110611
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Limit Theorems for Functionals of Random Convex Hulls in a Unit Disk

Isakjan Khamdamov ¹, Azam Imomov ^2,*
1 Department of Natural Sciences, Faculty of Economics, University of Public Safety of the Republic of Uzbekistan, Uzbekistan
² The Mathematics Department, Faculty of Mathematics and Computer Sciences, Karshi State University, Uzbekistan

ABSTRACT

In this article, we study the functionals of the convex hull generated by independent observations over two-dimensional random points. When the random points are given in the polar coordinate system, their components are independent of each other, the angular coordinate is distributed uniformly, and the tail of the distribution of the radial coordinate is a regularly varying function near the circle of the unit disk – support. Here, with the approximation of the binomial point process by an inhomogeneous Poisson one, it is possible to study the asymptotic properties of the main functionals of the convex hull. Using the independence property of the increment of Poisson processes, we find an asymptotic expression for the mean values and variances for the main functionals of the convex hull. Uniform boundedness of exponential moments is proved for the same functionals, in the case when the convex hull is generated from an inhomogeneous Poisson point process inside the disk. The indicated independence property of the increment of the Poisson process allows us to express the area of the convex hull as a sum of independent identically distributed random variables, with which we prove the central limiting theorem for the number of vertices and the area of the convex hull. From the results obtained, we can conclude that if the tail of the distribution near the boundary is heavier, then there are many elements of the sample near the support boundary, and this means that there are many vertices of the convex hull, but the area bounded by the perimeter of the convex hull and the circle, as well as the difference between the perimeter of the convex hull and the circle, becomes negligible.

KEYWORDS
Convex Hull, Poisson Point Process, Binomial Point Process

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Isakjan Khamdamov , Azam Imomov , "Limit Theorems for Functionals of Random Convex Hulls in a Unit Disk," Mathematics and Statistics, Vol. 11, No. 6, pp. 960 - 964, 2023. DOI: 10.13189/ms.2023.110611.

(b). APA Format:
Isakjan Khamdamov , Azam Imomov (2023). Limit Theorems for Functionals of Random Convex Hulls in a Unit Disk. Mathematics and Statistics, 11(6), 960 - 964. DOI: 10.13189/ms.2023.110611.