Normal Approximation of Poisson Functionals in Kolmogorov Distance
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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.
KeywordsCentral limit theorem Malliavin calculus Poisson point process Stein’s method U-statistic Wiener–Itô chaos expansion
Mathematics Subject ClassificationPrimary: 60F05 60H07 Secondary: 60G55
Large parts of this paper were written during a stay at Case Western Reserve University (February to July 2012) supported by the German Academic Exchange Service. The author thanks Elizabeth Meckes, Giovanni Peccati, Matthias Reitzner, and Christoph Thäle for some valuable hints and helpful discussions.
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