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Mean and Variance of Vacancy for Distribution of k-Dimensional Spheres within k-Dimensional Space

Peter Hall
Journal of Applied Probability
Vol. 21, No. 4 (Dec., 1984), pp. 738-752
DOI: 10.2307/3213692
Stable URL: http://www.jstor.org/stable/3213692
Page Count: 15
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Mean and Variance of Vacancy for Distribution of k-Dimensional Spheres within k-Dimensional Space
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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 $na_{n} \rightarrow 0, na_{n} \rightarrow a(0 < a < \infty)$ and nan → ∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

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