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Covering the circle with random arcs of random sizes

Published online by Cambridge University Press:  14 July 2016

Andrew F. Siegel  and
Lars Holst
Andrew F. Siegel*
Affiliation:
Princeton University
Lars Holst*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Statistics, Princeton University, Princeton, NJ 08544, U.S.A.
∗∗ Postal address: Department of Mathematics, Uppsala University, Thunbergsvägen 3, S-752 38 Uppsala, Sweden.
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Abstract

Consider the random uniform placement of a finite number of arcs on the circle, where the arc lengths are sampled from a distribution on (0, 1). We provide exact formulae for the probability that the circle is completely covered and for the distribution of the number of uncovered gaps, extending Stevens's (1939) formulae for the case of fixed equal arc lengths. A special class of arc length distributions is considered, and exact probabilities of coverage are tabulated for the uniform distribution on (0, 1). Some asymptotic results for the number of gaps are also given.

Keywords

GEOMETRICAL PROBABILITYCOVERAGE PROBABILITYSPACINGS
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 2 , June 1982 , pp. 373 - 381
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Research supported by the U.S. Army Research Office Grant DAAG29-79-C-0205, the U.S. Office of Naval Research Contract N00014-76-C-0475, a Fellowship from the American-Scandinavian Foundation, and a Grant from the Swedish Natural Science Research Council.

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