Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-09T07:10:21.259Z Has data issue: false hasContentIssue false
  • English
  • Français

Rates of Poisson convergence for some coverage and urn problems using coupling

Published online by Cambridge University Press:  14 July 2016

L. Holst ,
J. E. Kennedy  and
M. P. Quine
L. Holst*
Affiliation:
Uppsala University
J. E. Kennedy*
Affiliation:
Uppsala University
M. P. Quine*
Affiliation:
University of Sydney
*
∗Postal address: Department of Mathematics, Thunbergsv. 3, S-75238, Uppsala University, Sweden.
∗∗Postal address: Department of Mathematical Statistics, University of Sydney, NSW 2006, Australia.
∗∗Postal address: Department of Mathematical Statistics, University of Sydney, NSW 2006, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.

Keywords

VARIATION DISTANCESPACINGSCOUPLINGCIRCLE COVERAGEOCCUPANCY PROBLEMSMULTINOMIAL SAMPLINGPÓLYA SAMPLING
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 4 , December 1988 , pp. 717 - 724
Copyright
Copyright © Applied Probability Trust 1988 

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

Barbour, A. D. and Eagleson, G. K. (1983) Poisson approximation for some statistics based on exchangeable trials. Adv. Appl. Prob. 15, 585–600.Google Scholar
Barbour, A. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Barbour, A. D. and Holst, L. (1989) Some applications of the Stein–Chen method for proving Poisson convergence. Adv. Appl. Prob. To appear.Google Scholar
Brown, T. C. (1982) Poisson approximations and exchangeable random variables. In Exchangeability in Probability and Statistics , ed. Koch, G. and Spizzichino, F., North-Holland, Amsterdam, 177–183.Google 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
Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. 2, Wiley, New York.Google Scholar
Höglund, T. (1979) A unified formulation of the central limit theorem for small and large deviations from the mean. Z. Wahrscheinlichkeitsth. 49, 105–117.CrossRefGoogle Scholar
Holst, L. (1979) A unified approach to limit theorems for urn models. J. Appl. Prob. 16, 154–162.CrossRefGoogle Scholar
Holst, L. (1986) On birthday, collector's and other classical urn problems. Internat Statist. Rev. 54, 15–27.CrossRefGoogle Scholar
Holst, L. and Hüsler, J. (1984) On the random coverage of the circle. J. Appl. Prob. 21, 558–566.Google Scholar
Janson, S. (1986) Poisson convergence and Poisson processes with applications to random graphs. Uppsala Univ. Dept. Math. Report No. 1986:7 .Google Scholar
Kennedy, J. and Quine, M. (1986) Two applications of a Poisson approximation theorem. Manuscript, Univ. of Sydney.Google Scholar
Kolchin, V. F., Sevast'Yanov, B. A. and Chistyakov, V. P. (1978) Random Allocations. Winston, Washington DC.Google Scholar
Le Cam, L. (1960) An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 1181–1197.CrossRefGoogle Scholar
Sevast'Yanov, B. A. (1972) Poisson limit law for a scheme of sums of dependent random variables. Theory Prob. Appl. 17, 695–699.Google Scholar
Vatutin, V. A. and Mikhailov, V. G. (1982) Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles. Theory Prob. Appl. 27, 734–743.CrossRefGoogle Scholar
Wang, Y. H. (1986) Coupling methods in approximation. Canad. J. Statist. 14, 69–74.CrossRefGoogle Scholar