Open Access
August, 1985 On the Coverage of $k$-Dimensional Space by $k$-Dimensional Spheres
Peter Hall
Ann. Probab. 13(3): 991-1002 (August, 1985). DOI: 10.1214/aop/1176992920


Let $n k$-dimensional spheres, each of content $a_n$, be distributed within a $k$-dimensional cube according to density $f$. We derive necessary and sufficient conditions on $a_n$ in order that the probability that the cube is completely covered at least $\ell$ times by the spheres, tend to one as $n\rightarrow\infty$. (Here $\ell$ is an arbitrary positive integer.) In the special case $f\equiv$ const., we obtain upper and lower bounds of the same order of magnitude for the probability of incomplete coverage.


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Peter Hall. "On the Coverage of $k$-Dimensional Space by $k$-Dimensional Spheres." Ann. Probab. 13 (3) 991 - 1002, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0582.60015
MathSciNet: MR799434
Digital Object Identifier: 10.1214/aop/1176992920

Primary: 60E05

Keywords: coverage , geometric probability

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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