Normal Approximation of Poisson Functionals in Kolmogorov Distance

Matthias Schulte ()
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Matthias Schulte: Karlsruhe Institute of Technology

Journal of Theoretical Probability, 2016, vol. 29, issue 1, 96-117

Abstract: 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.

Keywords: Central limit theorem; Malliavin calculus; Poisson point process; Stein’s method; U-statistic; Wiener–Itô chaos expansion; Primary: 60F05; 60H07; Secondary: 60G55 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-014-0576-6

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