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A probabilistic proof of Stein's factors

Part of: Distribution theory Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

Aihua Xia
Aihua Xia*
Affiliation:
University of New South Wales
*
∗Postal address: School of Mathematics, The University of New South Wales, Kensington, NSW 2033, Australia. Email address: xia@maths.unsw.edu.au.
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Abstract

We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalizing puzzle in using Markov chain knowledge to view the celebrated Stein–Chen method for Poisson approximations. This work complements the work of Barbour (1988) for the case of Poisson random variable approximation.

Keywords

Poisson approximationStein–Chen methodbirth and death processcoupling

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E17: Approximations to distributions (nonasymptotic) 60J27: Continuous-time Markov processes on discrete state spaces
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 287 - 290
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This work was supported by an Australia Research Council Small Grant from the University of New South Wales.

References

Barbour, A. D. (1988). Stein's method and Poisson process convergence. In A Celebration of Applied Probability (J. Appl. Prob. 25A), ed. Gani, J. Applied Probability Trust, Sheffield, UK, pp. 175–184.Google Scholar
Barbour, A. D., and Brown, T. C. (1992). Stein's method and point process approximation. Stochast. Proc. Appl. 43, 9–31.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., Holst, L., and Janson, S. (1992). Poisson Approximation. OUP, Oxford, UK.Google Scholar
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar