The Annals of Probability

Covering Problems for Brownian Motion on Spheres

Peter Matthews

Full-text: Open access

Abstract

Bounds are given on the mean time taken by a strong Markov process to visit all of a finite collection of subsets of its state space. These bounds are specialized to Brownian motion on the surface of the unit sphere $\Sigma_p$ in $R^p$. This leads to bounds on the mean time taken by this Brownian motion to come within a distance $\varepsilon$ of every point on the sphere and bounds on the mean time taken to come within $\varepsilon$ of every point or its opposite. The second case is related to the Grand Tour, a technique of multivariate data analysis that involves a search of low-dimensional projections. In both cases the bounds are asymptotically tight as $\varepsilon \rightarrow 0$ on $\Sigma_p$ for $p \geq 4$.

Article information

Source
Ann. Probab., Volume 16, Number 1 (1988), 189-199.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991894

Digital Object Identifier
doi:10.1214/aop/1176991894

Mathematical Reviews number (MathSciNet)
MR920264

Zentralblatt MATH identifier
0638.60014

JSTOR
links.jstor.org

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G17: Sample path properties 60E15: Inequalities; stochastic orderings 58G32

Keywords
Brownian motion Grand Tour hitting time sphere covering rapid mixing

Citation

Matthews, Peter. Covering Problems for Brownian Motion on Spheres. Ann. Probab. 16 (1988), no. 1, 189--199. doi:10.1214/aop/1176991894. https://projecteuclid.org/euclid.aop/1176991894


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