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Normal Approximation of Poisson Functionals in Kolmogorov Distance

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  • Matthias Schulte

    (Karlsruhe Institute of Technology)

Abstract

Peccati, Solè, Taqqu, and Utzet recently combined Stein’s method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper was to show this behavior for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.

Suggested Citation

  • Matthias Schulte, 2016. "Normal Approximation of Poisson Functionals in Kolmogorov Distance," Journal of Theoretical Probability, Springer, vol. 29(1), pages 96-117, March.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:1:d:10.1007_s10959-014-0576-6
    DOI: 10.1007/s10959-014-0576-6
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    References listed on IDEAS

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    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.
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    Cited by:

    1. Lo, Tiffany Y.Y. & Xia, Aihua, 2024. "On the rate of normal approximation for Poisson continuum percolation," Statistics & Probability Letters, Elsevier, vol. 210(C).
    2. Betken, Carina & Hug, Daniel & Thäle, Christoph, 2023. "Intersections of Poisson k-flats in constant curvature spaces," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 96-129.

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