Annals of Applied Probability

Upper bounds for spatial point process approximations

Dominic Schuhmacher

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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.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 615-651.

First available in Project Euclid: 1 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 62E20: Asymptotic distribution theory 62G07: Density estimation

Point processes Poisson process approximation Stein’s method density estimation total variation distance dt₂-distance


Schuhmacher, Dominic. Upper bounds for spatial point process approximations. Ann. Appl. Probab. 15 (2005), no. 1B, 615--651. doi:10.1214/105051604000000684.

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