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.
Publisher
Cambridge University Press (CUP)
Subject
Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability
Cited by
3 articles.
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1. Peter Gavin Hall. 20 November 1951—9 January 2016;Biographical Memoirs of Fellows of the Royal Society;2018-01-31
2. Peter Gavin Hall 1951–2016;Historical Records of Australian Science;2017
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