Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T21:14:46.182Z Has data issue: false hasContentIssue false
  • English
  • Français

On point processes on the circle

Published online by Cambridge University Press:  14 July 2016

Jürg Hüsler
Jürg Hüsler*
Affiliation:
University of Berne
*
∗ Postal address: Dept. of Mathematical Statistics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland.
Get access Rights & Permissions [Opens in a new window]

Abstract

Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.

Keywords

COVERAGEVACANCYEXTREME GAPSRANDOM ARCSCONVERGENCE

Information

Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 322 - 331
Copyright
Copyright © Applied Probability Trust 1986 

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.)

Article purchase

Temporarily unavailable

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
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Holst, L. (1980) 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, 643–655.Google Scholar
Holst, L. and Hüsler, J. (1984) On the random coverage of the circle. J. Appl. Prob. 21, 558–566.Google Scholar
Hüsler, J. (1982a) On the random coverage of the circle and asymptotic distributions. J. Appl. Prob. 19, 578–587.Google Scholar
Hüsler, J. (1982b) On the maximal covering of the circle. Technical Report, Dept. of Math. Statistics, University of Bern.Google Scholar
Hüsler, J. (1983) A note on random coverage of the circle. Proc. Conf. Stochastic Geometry, Geometric Stochastic, Oberwolfach 1983. Teubner, Leipzig.Google Scholar
Kallenberg, O. (1976) Random Measures. Akademie Verlag, Berlin.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. (1979) Asymptotic coverage distributions on the circle. Ann. Prob. 7, 651–661.CrossRefGoogle Scholar
Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia.Google Scholar