IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v43y1992i1p9-31.html
   My bibliography  Save this article

Stein's method and point process approximation

Author

Listed:
  • Barbour, A. D.
  • Brown, T. C.

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.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:43:y:1992:i:1:p:9-31
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(92)90073-Y
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Phelan, Michael J., 1997. "Approach to stationarity for birth and death on flows," Stochastic Processes and their Applications, Elsevier, vol. 66(2), pages 183-207, March.
    2. 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.
    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.
    4. Brown, Timothy C. & Xia, Aihua, 1995. "On Stein-Chen factors for Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 23(4), pages 327-332, June.
    5. Schuhmacher, Dominic, 2005. "Distance estimates for dependent superpositions of point processes," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1819-1837, November.
    6. Schulte, Matthias & Thäle, Christoph, 2012. "The scaling limit of Poisson-driven order statistics with applications in geometric probability," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 4096-4120.
    7. He, Shengwu & Xia, Aihua, 1997. "On poisson approximation to the partial sum process of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 68(1), pages 101-111, May.
    8. Gan, H.L. & Xia, A., 2015. "Stein’s method for conditional compound Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 19-26.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:43:y:1992:i:1:p:9-31. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.