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Upper bounds for spatial point process approximations


Schuhmacher, D (2005). Upper bounds for spatial point process approximations. Annals of Applied Probability, 15(1B):615-651.

Abstract

We consider the behavior of spatial point processes when subjected to a class of linear transformations indexed by a variable T. It was shown in Ellis [Adv. in Appl. Probab. 18 (1986) 646–659] that, under mild assumptions, the transformed processes behave approximately like Poisson processes for large T. In this article, under very similar assumptions, explicit upper bounds are given for the d2-distance between the corresponding point process distributions. A number of related results, and applications to kernel density estimation and long range dependence testing are also presented. The main results are proved by applying a generalized Stein–Chen method to discretized versions of the point processes.

Abstract

We consider the behavior of spatial point processes when subjected to a class of linear transformations indexed by a variable T. It was shown in Ellis [Adv. in Appl. Probab. 18 (1986) 646–659] that, under mild assumptions, the transformed processes behave approximately like Poisson processes for large T. In this article, under very similar assumptions, explicit upper bounds are given for the d2-distance between the corresponding point process distributions. A number of related results, and applications to kernel density estimation and long range dependence testing are also presented. The main results are proved by applying a generalized Stein–Chen method to discretized versions of the point processes.

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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
Uncontrolled Keywords:Point processes, Poisson process approximation, Stein’s method, density estimation, total variation distance, dt₂-distance
Language:English
Date:2005
Deposited On:29 Nov 2010 16:26
Last Modified:20 Feb 2018 11:42
Publisher:Institute of Mathematical Statistics
ISSN:1050-5164
OA Status:Hybrid
Publisher DOI:https://doi.org/10.1214/105051604000000684
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1067.60022
http://www.ams.org/mathscinet-getitem?mr=2114984

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