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Distance estimates for dependent superpositions of point processes

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  • Schuhmacher, Dominic

Abstract

In this article, superpositions of possibly dependent point processes on a general space are considered. Using Stein's method for Poisson process approximation, an estimate is given for the Wasserstein distance d2 between the distribution of such a superposition and an appropriate Poisson process distribution. This estimate is compared to a modern version of Grigelionis' theorem, and to results of Banys [Lecture Notes in Statistics, vol. 2, Springer, New York, 1980, pp. 26-37], Arratia et al. [Ann. Probab. 17 (1989) 9-25] and Barbour et al. [Poisson Approximation, Oxford University Press, Oxford, 1992]. Furthermore, an application to a spatial birth-death model is presented.

Suggested Citation

  • Schuhmacher, Dominic, 2005. "Distance estimates for dependent superpositions of point processes," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1819-1837, November.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:11:p:1819-1837
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    References listed on IDEAS

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    1. Barbour, A. D. & Brown, T. C., 1992. "Stein's method and point process approximation," Stochastic Processes and their Applications, Elsevier, vol. 43(1), pages 9-31, November.
    2. Richard Serfozo, 1984. "Thinning of Cluster Processes: Convergence of Sums of Thinned Point Processes," Mathematics of Operations Research, INFORMS, vol. 9(4), pages 522-533, November.
    3. Brown, Timothy C. & Weinberg, Graham V. & Xia, Aihua, 2000. "Removing logarithms from Poisson process error bounds," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 149-165, May.
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    Cited by:

    1. Cong, Tianshu & Xia, Aihua & Zhang, Fuxi, 2020. "A large sample property in approximating the superposition of i.i.d. finite point processes," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4493-4511.
    2. Pianoforte, Federico & Schulte, Matthias, 2022. "Criteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellations," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 388-422.
    3. Xia, Aihua & Zhang, Fuxi, 2008. "A polynomial birth-death point process approximation to the Bernoulli process," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1254-1263, July.

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