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Mean and variance of vacancy for distribution of k-dimensional spheres within k-dimensional space

Published online by Cambridge University Press:  14 July 2016

Peter Hall
Peter Hall*
Affiliation:
The Australian National University
*
*Postal address: Department of Statistics, Faculty of Economics, The Australian National University, GPO Box 4, ACT 2601, Australia.
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Abstract

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an → 0. The formulae separate naturally into three cases, corresponding to nan → 0, nan → a (0 < a <∞) and nan →∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

Keywords

COVERAGEGEOMETRIC PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 4 , December 1984 , pp. 738 - 752
Copyright
Copyright © Applied Probability Trust 1984 

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