IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v118y2008i7p1254-1263.html
   My bibliography  Save this article

A polynomial birth-death point process approximation to the Bernoulli process

Author

Listed:
  • Xia, Aihua
  • Zhang, Fuxi

Abstract

We propose a class of polynomial birth-death point processes (abbreviated to PBDP) , where Z is a polynomial birth-death random variable defined in [T.C. Brown, A. Xia, Stein's method and birth-death processes, Ann. Probab. 29 (2001) 1373-1403], Ui's are independent and identically distributed random elements on a compact metric space, and Ui's are independent of Z. We show that, with two appropriately chosen parameters, the error of PBDP approximation to a Bernoulli process is of the order O(n-1/2) with n being the number of trials in the Bernoulli process. Our result improves the performance of Poisson process approximation, where the accuracy is mainly determined by the rarity (i.e. the success probability) of the Bernoulli trials and the dependence on sample size n is often not explicit in the bound.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:7:p:1254-1263
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(07)00141-X
    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.

    References listed on IDEAS

    as
    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. Schuhmacher, Dominic, 2005. "Distance estimates for dependent superpositions of point processes," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1819-1837, 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.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Schuhmacher, Dominic, 2005. "Distance estimates for dependent superpositions of point processes," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1819-1837, November.
    2. 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.
    3. 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.
    4. 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.
    5. Gan, H.L. & Xia, A., 2015. "Stein’s method for conditional compound Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 19-26.
    6. 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.
    7. Brown, Timothy C. & Xia, Aihua, 1995. "On Stein-Chen factors for Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 23(4), pages 327-332, June.
    8. 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.
    9. 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.

    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:118:y:2008:i:7:p:1254-1263. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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.