Convex hulls of uniform samples from a convex polygon
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.60011
author: Piet Groeneboom
published in: Advances in Applied Probability
publication date: 2012-07-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.3118
full work available at URL: https://projecteuclid.org/euclid.aap/1339878714
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 6055124
zbMATH Keywords: convex hull
zbMATH Keywords: convex polygon
zbMATH Keywords: Poisson point process
MaRDI profile type: Publication
OpenAlex ID: W2963873837
arXiv ID: 1109.3118
cites work: Poisson polytopes
cites work: On the variance of random polytopes
cites work: Q3410253
cites work: An identity relating moments of functionals of convex hulls
cites work: On the boundary structure of the convex hull of random points
cites work: Limit theorems for functionals of convex hulls
cites work: The concave majorant of Brownian motion
cites work: Limit theorems for convex hulls
cites work: Vertices of the least concave majorant of Brownian motion with parabolic drift
cites work: Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain
DOI: 10.1239/AAP/1339878714
title: Convex hulls of uniform samples from a convex polygon
review text: Let \(N_n\) be the number of vertices of the convex hull of a sample of size \(n\), drawn uniformly from the interior of a convex polygon with \(r\) vertices. It was shown in [\textit{P. Groeneboom}, Probab. Theory Relat. Fields 79, No. 3, 327--368 (1988; Zbl 0635.60012)] that \(\{N_n -{{2r}\over{3}} \log n\}/\{{{10r}\over{27}}\log n\}^{1/2}\to {\mathcal N}(0, 1)\) in distribution, where \({\mathcal N}(0, 1)\) denotes the standard normal distribution. The central limit result for \(N_n\) was subsequently derived from a corresponding result for the boundary of the convex hull of the approximating Poisson point process. \textit{A. V. Nagaev} and \textit{I. M. Khamdanov} extended in an unpublished paper [``Limit theorems for functionals of random convex hulls'' (Russian), Preprint. Institute of Mathematics, Academy of Sciences of Uzbekistan] the central limit theorem for the number of verticies \(N_n\) from [P. Groeneboom, loc. cit.]. NEWLINENEWLINEThe aim of this paper to give a simple proof of the result from [A. V. Nagaev and I. M. Khamdanov, loc. cit.].
reviewed by: Viktor Ohanyan
description: scientific article; zbMATH DE number 6055124