Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-26T20:55:10.186Z Has data issue: false hasContentIssue false
  • English
  • Français

On the random coverage of the circle

Published online by Cambridge University Press:  14 July 2016

L. Holst  and
J. Hüsler
L. Holst*
Affiliation:
Uppsala University
J. Hüsler*
Affiliation:
University of Bern
*
∗Postal address: Department of Mathematics, Uppsala University, Thunbergsvägen 3, S-752 38 Uppsala, Sweden.
∗∗Postal address: Department of Mathematical Statistics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland.
Get access Rights & Permissions [Opens in a new window]

Abstract

Place n arcs of equal length a uniformly at random on the circumference of a circle. We discuss the joint limit distributions of the number of gaps, the uncovered proportion of the circle and the lengths of the largest gap and of the smallest gap, depending on how a → 0 as n →∞.

We show that the results may be proved in a unified and simple way by using a result of Le Cam.

Keywords

JOINT LIMIT THEOREMSCOVERING A CIRCLESPACINGSUNIFORM DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 3 , September 1984 , pp. 558 - 566
Copyright
Copyright © Applied Probability Trust 1984 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Darling, D. A. (1953) On a class of problems related to the random division of an interval. Ann. Math. Statist. 24, 239–253.Google Scholar
Edens, E. (1975) Random covering of a circle. Indag. Math. 37, 373–384.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Flatto, L. (1973) A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184.CrossRefGoogle Scholar
Flatto, L. and Konheim, A. (1962) The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.Google Scholar
Holst, L. (1980a) On multiple covering of a circle with random arcs. J. Appl. Prob. 17, 284–290.Google Scholar
Holst, L. (1980b) On the lengths of the pieces of a stick broken at random. J. Appl. Prob. 17, 623–634.Google Scholar
Holst, L. (1981) On convergence of the coverage by random arcs on a circle and the largest spacing. Ann. Prob. 9, 648–655.CrossRefGoogle Scholar
Hüsler, J. (1982a) Random coverage of the circle and asymptotic distributions. J. Appl. Prob. 19, 578–587.Google Scholar
Hüsler, J. (1982b) A note on higher order spacings. Technical Report, Department of Mathematical Statistics, University of Bern.Google Scholar
Kaplan, N. (1977) Two applications of a Poisson approximation for dependent events. Ann. Prob. 5, 787–794.Google Scholar
Kaplan, N. (1978) A limit theorem for random coverings of a circle which do not quite cover. J. Appl. Prob. 15, 443–446.Google Scholar
Le Cam, L. (1958) Un théorème sur la division d'un intervalle par des points pris au hasard. Publ. Inst. Statist. Univ. Paris 7, 7–16.Google Scholar
Levy, P. (1939) Sur la division d'un segment par des points choisis au hasard. C.R. Acad. Sci. Paris 208, 147–149.Google Scholar
Siegel, A. F. (1978) Random coverage of the circle. J. Appl. Prob. 15, 774–789.Google Scholar
Siegel, A. F. (1979) Asymptotic coverage distributions on the circle. Ann. Prob. 7, 651–661.Google Scholar
Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia.Google Scholar
Steutel, F. W. (1967) Random division of an interval. Statist. Neerlandica 21, 231–244.Google Scholar
Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics, London 9, 315–320.CrossRefGoogle Scholar