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Random space filling and moments of coverage in geometrical probability

Published online by Cambridge University Press:  14 July 2016

Andrew F. Siegel
Andrew F. Siegel*
Affiliation:
Stanford University
*
∗Now at the University of Wisconsin-Madison.
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Abstract

The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases.

Keywords

GEOMETRICAL PROBABILITYCOVERAGE SPACEMOMENTS OF COVERAGECOVERAGE PROBABILITYMONTE CARLORANDOM ARCS ON THE CIRCLE
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 340 - 355
Copyright
Copyright © Applied Probability Trust 1978 

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Footnotes

This research forms part of the author's Ph.D. dissertation, done at Stanford University under Professor Herbert Solomon and was supported by the U.S. Army Research Office. Grant No. DAAG29–77–G–0031, the Office of Naval Research, Contract No. N00014–76–C–0475, and the Public Health Service Training Grant No. 5–T01–GM00025.

∗

Shepp (1972) notes this in the case in which we condition on the arc lengths.

References

Ailam, G. (1966) Moments of coverage and coverage spaces. J. Appl. Prob. 3, 550–555.CrossRefGoogle Scholar
Birnbaum, Z. W. (1948) On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81.CrossRefGoogle Scholar
Gilbert, E. N. (1965) The probability of covering a sphere with N circular caps. Biometrika 52, 323–330.CrossRefGoogle Scholar
Moran, P. A. P. and Fazekas De St. Groth, S. (1962) Random circles on a sphere. Biometrika 49, 389–396.CrossRefGoogle Scholar
Robbins, H. E. (1944) On the measure of a random set. Ann. Math. Statist. 15, 70–74.CrossRefGoogle Scholar
Shepp, L. A. (1972) Covering the circle with random arcs. Israel J. Maths 11, 328–345.CrossRefGoogle Scholar
Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugen. 9, 315–320.CrossRefGoogle Scholar