Three limit theorems for vacancy in multivariate coverage problems
type: Item
instance of: scholarly article
title: Three limit theorems for vacancy in multivariate coverage problems
zbMATH Open document ID: 0602.60016
DOI: 10.1016/0047-259X(85)90034-X
published in: Journal of Multivariate Analysis
publication date: 1985-01-01T00:00:00Z
review text: The three limit theorems of the title concern the asymptotic distribution of V, the volume left uncovered when a large number n of small spheres (each of volume \(a_ n)\) are placed at random (according to a fairly general distribution) in a unit cube in \({\mathbb{R}}^ k\). The first two theorems establish asymptotic normality of V when the sum of the sphere volumes \(na_ n\) either tends to a positive constant or tends to zero but not too fast. The third establishes a compound Poisson limit of a standardised version of V when \(n^ 2a_ n\) tends to zero. The method of proof for the first two theorems involves dissection of the cube into subcubes, moment estimates on the vacancy within slightly shrunk subcubes, and the Berry-Esseen theorem. For the third theorem estimates are made of the number of intersecting spheres.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3971867
zbMATH Keywords: central limit theorem
zbMATH Keywords: geometric probability
zbMATH Keywords: vacancy
zbMATH Keywords: compound Poisson limit
zbMATH Keywords: Berry-Esseen theorem
reviewed by: Wilfrid S. Kendall
MaRDI profile type: Publication
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full work available at URL: 0047-259x(85)90034-x
OpenAlex ID: W2091859301
author: Peter Hall
description: scientific article; zbMATH DE number 3971867