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August, 1979 Asymptotic Coverage Distributions on the Circle
Andrew F. Siegel
Ann. Probab. 7(4): 651-661 (August, 1979). DOI: 10.1214/aop/1176994988


Place $n$ arcs, each of length $a_n$, uniformly at random on the circumference of a circle, choosing the arc length sequence $a_n$ so that the probability of completely covering the circle remains constant. We obtain the limiting distribution of the uncovered proportion of the circle. We show that this distribution has a natural interpretation as a noncentral chi-square distribution with zero degrees of freedom by expressing it as a Poisson mixture of mass at zero with central chi-square deviates having even degrees of freedom. We also treat the case of proportionately smaller arcs and obtain a limiting normal distribution. Potential applications include immunology, genetics, and time series analysis.


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Andrew F. Siegel. "Asymptotic Coverage Distributions on the Circle." Ann. Probab. 7 (4) 651 - 661, August, 1979.


Published: August, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0412.60014
MathSciNet: MR537212
Digital Object Identifier: 10.1214/aop/1176994988

Primary: 60D05
Secondary: 60E05 , 60F99 , 62F99

Keywords: Asymptotic coverage distribution , Geometrical probability , noncentral chi-square with zero degrees of freedom , random arcs

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 4 • August, 1979
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