Covering problems for Brownian motion on spheres
type: Item
instance of: scholarly article
title: Covering problems for Brownian motion on spheres
zbMATH Open document ID: 0638.60014
author: B. George
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: Consider Brownian motion on the unit sphere \(\Sigma_ p\) and \({\mathbb{R}}^ p\), \(p\geq 3\), and let \(C_ 1(\epsilon,p)\) be the first time a Brownian path on \(\Sigma_ p\) has come within distance \(\epsilon\) of all points of \(\Sigma_ p\). Similarly, \(C_ 2(\epsilon,p)\) is defined if points on \(\Sigma_ p\) are replaced by antipodal pairs. For the mean times \(EC_ 1(\epsilon,p)\) and \(EC_ 2(\epsilon,p)\), asymptotic bounds are given (as \(\epsilon\to 0)\) which are tight for \(p\geq 4\). The bounds involve the expected hitting times of caps on \(\Sigma_ p\) and the number of caps needed to cover \(\Sigma_ p\). Their proof is based on a related more general result for strong Markov processes. The problem is motivated by a Grand Tour, a technique in multivariate data analysis.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 4040923
zbMATH Keywords: hitting time
zbMATH Keywords: sphere covering
zbMATH Keywords: rapid mixing
zbMATH Keywords: strong Markov processes
zbMATH Keywords: Grand Tour
reviewed by: Wolfgang Weil
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991894
OpenAlex ID: W1986507557
DOI: 10.1214/AOP/1176991894
description: scientific article; zbMATH DE number 4040923