The limit distribution of the perimeter of a convex hull generated by a Poisson point process in a convex polygon

Abstract

In this paper, we study various functionals of convex hulls generated by the realization of a homogeneous Poisson point process in a polygon on a plane. The convex hull is a generalization of the extreme elements of the sample when estimating the carrier of the distribution; and in the multidimensional case, as an estimate of the carrier of the distribution, it retains many properties of one-dimensional estimates, such as consistency, asymptotic unbiasedness, and sufficiency. Works on the study of random convex hulls in polygons and various functionals of them are usually referred to the field of probabilistic geometry. It should be noted that studying the properties of even the simplest functionals of convex hulls, such as the number of vertices or the area, is not an easy problem (see, for example, [1-4]). This also explains the fact that before the appearance of the work of P. Groeneboom [6], the main progress in this field was achieved only in the study of the properties of the mean values of such functionals. In [6], he succeeded in proving the central limit theorem for the number of vertices of a convex hull in the case when the support of the original uniform distribution is either a convex polygon or an ellipse. The main result of this paper consists in proving that the difference between the perimeters of the distribution carrier and the convex hull converges in probability to a random variable that has a distribution different from normal, and it is asymptotically independent of the number of vertices and the area of the convex hull.