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January, 1988 Covering Problems for Brownian Motion on Spheres
Peter Matthews
Ann. Probab. 16(1): 189-199 (January, 1988). DOI: 10.1214/aop/1176991894


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 $p \geq 4$.


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Peter Matthews. "Covering Problems for Brownian Motion on Spheres." Ann. Probab. 16 (1) 189 - 199, January, 1988.


Published: January, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0638.60014
MathSciNet: MR920264
Digital Object Identifier: 10.1214/aop/1176991894

Primary: 60D05
Secondary: 58G32 , 60E15 , 60G17

Keywords: Brownian motion , Grand Tour , hitting time , rapid mixing , sphere covering

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 1 • January, 1988
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