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Some applications of the Stein-Chen method for proving Poisson convergence

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour and
Lars Holst
A. D. Barbour*
Affiliation:
Universität Zürich
Lars Holst*
Affiliation:
Uppsala University
*
∗ Postal address: Institut für Angewandte Mathematik, Universität Zürich, Rämistrasse 74, CH-8001 Zürich, Switzerland.
∗∗ Postal address: Uppsala University, Department of Mathematics, Thunbergsv. 3, S-752 38 Uppsala, Sweden.
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Abstract

Let W be a sum of Bernoulli random variables and U λ a Poisson random variable having the same mean λ = EW. Using the Stein-Chen method and suitable couplings, general upper bounds for the variational distance between W and U λ are given. These bounds are applied to problems of occupancy, using sampling with and without replacement and Pólya sampling, of capture-recapture, of spacings and of matching and ménage.

Keywords

VARIATIONAL DISTANCECOUPLINGOCCUPANCYCAPTURE–RECAPTURESPACINGSMATCHING

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 1 , March 1989, pp. 74 - 90
Copyright
Copyright © Applied Probability Trust 1989 

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References

Arratia, R., Goldstein, L. and Gordon, L. (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17 (January).Google Scholar
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. D. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.Google Scholar
Bogart, K. P. and Doyle, P. (1986) Non-sexist solution of the ménage problem. Amer. Math. Monthly. 93, 514–518.Google Scholar
Chen, L. H. Y. (1975a) Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Chen, L. H. Y. (1975b) An approximation theorem for sums of certain randomly selected indicators. Z. Wahrscheinlichkeitsth. 33, 69–74.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Holst, L. (1980) On matrix occupancy, commitee, and capture-recapture problems. Scand. J. Statist. 7, 139–146.Google Scholar
Holst, L. (1986) On birthday, collectors’, occupancy and other classical urn problems. Int. Statist. Rev. 54, 15–27.Google Scholar
Holst, L. and Hüsler, J. (1984) On the random coverage of the circle. J. Appl. Prob. 21, 558–566.Google Scholar
Holst, L., Kennedy, J. and Quine, ?. (1988) Rates of Poisson convergence for some coverage and urn problems using coupling. J. Appl. Prob. 25, 717–724.Google Scholar
Knox, G. (1964) Epidemiology of childhood leukaemia in Northumberland and Durham. Brit. J. Prev. Soc. Med. 18, 17–24.Google Scholar
Kolchin, V. F., Sevast’Yanov, B. A. and Chistyakov, V. P. (1978) Random Allocations. Winston, Washington DC.Google Scholar
Lanke, J. (1973) Asymptotic results on matching distributions. J. R. Statist. Soc. B. 35, 117–122.Google 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
Stein, C. (1970) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 583–602.Google Scholar
Stein, C. (1986) Approximate Computation of Expectations. Inst. Math. Statist. Lecture Notes–Monograph Series Vol. 7, Hayward, California.Google Scholar
Takács, L. (1981) On the ‘Problème des Ménages’. Discrete Math. 36, 289–297.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.Google Scholar