Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T12:41:34.526Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-random limit for the volume covered k times

Published online by Cambridge University Press:  14 July 2016

A. J. Stam
A. J. Stam*
Affiliation:
Rijksuniversiteit Groningen
*
∗Postal address: Mathematisch Instituut, Rijksuniversiteit te Groningen, Postbus 800, Groningen, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let n sets with volume ~ n-–1 be placed in Rm, independently and with the same distribution. As n →∞, the volume in V CRm, covered by exactly k of these sets under certain conditions converges to a non-random limit, which is the integral over V of a density that is of the type of a Poisson probability.

Keywords

RANDOM SETSVOLUME COVEREDCONVERGENCE IN MEAN
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 148 - 156
Copyright
Copyright © Applied Probability Trust 

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

Ailam, G. (1966) Moments of coverage and coverage spaces. J. Appl. Prob. 3, 550–555.Google Scholar
Ailam, G. (1968) On probability properties of measures of random sets and the asymptotic behaviour of empirical distribution functions. J. Appl. Prob. 5, 196–202.Google Scholar
Ailam, G. (1970) The asymptotic distribution of the measure of random sets with application to the classical occupancy problem and suggestions for curve fitting. Ann. Math. Statist. 41, 427–439.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Kendall, D. G. (1974) Foundations of a theory of random sets. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G. Wiley, New York.Google Scholar
Le Cam, L. (1960) An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 1181–1197.Google Scholar
Moran, P. A. P. (1974) The volume occupied by normally distributed spheres. Acta Math. 133, 273–286.Google Scholar
Nicholson, A. J. (1933) The balance of animal populations. J. Animal Ecology 2, 132–178.Google Scholar
Rudin, W. (1974) Real and Complex Analysis, 2nd edn. McGraw-Hill, New York.Google Scholar
Siegel, A. F. (1979) Asymptotic coverage distributions on the circle. Ann. Prob. 7, 651–661.Google Scholar