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The scaling limit of Poisson-driven order statistics with applications in geometric probability

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  • Schulte, Matthias
  • Thäle, Christoph

Abstract

Let ηt be a Poisson point process of intensity t≥1 on some state space Y and let f be a non-negative symmetric function on Yk for some k≥1. Applying f to all k-tuples of distinct points of ηt generates a point process ξt on the positive real half-axis. The scaling limit of ξt as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m-th smallest point of ξt is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:12:p:4096-4120
    DOI: 10.1016/j.spa.2012.08.011
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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. Henze, Norbert & Klein, Timo, 1996. "The Limit Distribution of the Largest Interpoint Distance from a Symmetric Kotz Sample," Journal of Multivariate Analysis, Elsevier, vol. 57(2), pages 228-239, May.
    3. Janson, Svante, 1987. "Poisson convergence and poisson processes with applications to random graphs," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 1-30.
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    Cited by:

    1. Lachièze-Rey, Raphaël & Peccati, Giovanni, 2013. "Fine Gaussian fluctuations on the Poisson space II: Rescaled kernels, marked processes and geometric U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4186-4218.
    2. Matthias Schulte & Christoph Thäle, 2014. "Distances Between Poisson k -Flats," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 311-329, June.
    3. Owada, Takashi, 2019. "Topological crackle of heavy-tailed moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 4965-4997.
    4. Chenavier, Nicolas, 2014. "A general study of extremes of stationary tessellations with examples," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2917-2953.

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