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Stein's method and poisson process convergence

Published online by Cambridge University Press:  14 July 2016

A. D. Barbour
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Abstract

Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.

Keywords

RATE OF CONVERGENCE: POISSON POINT PROCESS
Type
Part 5 - Concepts of Coincidence and Convergence
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988 , pp. 175 - 184
Copyright
Copyright © Applied Probability Trust 1988 

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References

Barbour, A. D. (1980) Equilibrium distributions for Markov population processes. Adv. Appl. Prob. 12, 591–614.CrossRefGoogle Scholar
Barbour, A. D. (1982) Poisson convergence and random graphs. Math. Proc. Camb. Phil. Soc. 92, 349–359.CrossRefGoogle Scholar
Barbour, A. D. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Chen, L. H. Y. (1975) Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.CrossRefGoogle Scholar
Chen, L. H. Y. (1978) Two central limit problems for dependent random variables. Z. Wahrscheinlichkeitsth. 43, 223–243.CrossRefGoogle Scholar
Chen, T. H. Y. (1987) The rate of convergence in a central limit theorem for dependent random variables with arbitrary index set. Ann. Prob. Google Scholar
Karr, A. F. and Serfling, R. J. (1987) Poisson approximation of Bernoulli point processes and their superpositions, via coupling.Google Scholar
Michel, R. (1987) An improved error bound for the compound Poisson approximation. ASTIN Bull. 17.Google Scholar
Pitman, J. W. (1974) Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrscheinlichkeitsth. 29, 193–227.CrossRefGoogle 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
Takahata, H. (1983) On the rates in the central limit theorem for weakly dependent random fields. Z. Wahrscheinlichkeitsth. 64, 445–456.CrossRefGoogle Scholar