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Stein's method and point process approximation


Barbour, A D; Brown, T (1992). Stein's method and point process approximation. Stochastic Processes and their Applications, 43(1):9-31.

Abstract

The Stein-Chen method for Poisson approximation is adapted into a form suitable for obtaining error estimates for the approximation of the whole distribution of a point process on a suitable topological space by that of a Poisson process. The adaptation involves consideration of an immigration-death process on the topological space, whose equilibrium distribution is that of the approximating Poisson process; the Stein equation has a simple interpretation in terms of the generator of the immigration-death process. The error estimates for process approximation in total variation do not have the ‘magic’ Stein-Chein multiplying constants, which for univariate approximation tend to zero as the mean gets larger, but examples, including Bernoulli trials and the hard-core model on the torus, show that this is not possible. By choosing weaker metrics on the space of distributions of point processes, it is possible to reintroduce these constants. The proofs actually yield an improved estimate for one of the constants in the univariate case.

The Stein-Chen method for Poisson approximation is adapted into a form suitable for obtaining error estimates for the approximation of the whole distribution of a point process on a suitable topological space by that of a Poisson process. The adaptation involves consideration of an immigration-death process on the topological space, whose equilibrium distribution is that of the approximating Poisson process; the Stein equation has a simple interpretation in terms of the generator of the immigration-death process. The error estimates for process approximation in total variation do not have the ‘magic’ Stein-Chein multiplying constants, which for univariate approximation tend to zero as the mean gets larger, but examples, including Bernoulli trials and the hard-core model on the torus, show that this is not possible. By choosing weaker metrics on the space of distributions of point processes, it is possible to reintroduce these constants. The proofs actually yield an improved estimate for one of the constants in the univariate case.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1992
Deposited On:04 Nov 2009 14:37
Last Modified:05 Apr 2016 13:28
Publisher:Elsevier
ISSN:0304-4149
Free access at:Related URL. An embargo period may apply.
Publisher DOI:10.1016/0304-4149(92)90073-Y
Related URLs:http://user.math.uzh.ch/barbour/pub/Barbour/BBrown_PointProcess.pdf

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