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

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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
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    References listed on IDEAS

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    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. 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.
    3. 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.
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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.

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