Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T22:53:11.734Z Has data issue: false hasContentIssue false
  • English
  • Français

On multiple covering of a circle with random arcs

Published online by Cambridge University Press:  14 July 2016

Lars Holst
Lars Holst*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University. Thunbergsvägen 3, S–75238 Uppsala, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

On a circle of unit circumference arcs of length a are placed at random. Let Nα be equal to the necessary number of arcs to cover at least the length 1 − p, 0 ≦ p < 1, of the circumference at least m (≧1) times. In the present paper limit distributions of Nα are derived when α → 0. Some results for spacings are also obtained.

Keywords

COVERING A CIRCLESPACINGSUNIFORM DISTRIBUTIONLIMIT THEOREMS
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 284 - 290
Copyright
Copyright © Applied Probability Trust 1980 

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

Cressie, N. (1977) The minimum of higher-order gaps. Austral. J. Statist. 19, 132–143.CrossRefGoogle Scholar
Darling, D. A. (1953) On a class of problems related to the random division of an interval. Ann. Math. Statist. 24, 239–253.CrossRefGoogle Scholar
Edens, E. (1975) Random covering of a circle. Indag. Math. 37, 373–384.CrossRefGoogle Scholar
Flatto, L. (1973) A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184.CrossRefGoogle Scholar
Flatto, L. and Newman, D. J. (1977) Random coverings. Acta Math. 138, 241–264.CrossRefGoogle Scholar
Holst, L. (1979) Asymptotic normality of sum-functions of spacings. Ann. Prob. CrossRefGoogle Scholar
Kaplan, N. (1977) Two applications of a Poisson approximation for dependent events. Ann. Prob. 5, 787–794.CrossRefGoogle Scholar
Kaplan, N. (1978) A limit theorem for random coverings of a circle which do not quite cover. J. Appl. Prob. 15, 443–446.CrossRefGoogle Scholar
Leadbetter, M. R. (1974) On extreme values in stationary sequences. Z. Wahrscheinlichkeitsth. 28, 289–303.CrossRefGoogle Scholar
Lecam, 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
Shepp, L. A. (1972) Covering the circle with random arcs. Israel J. Math. 11, 328–345.CrossRefGoogle Scholar