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Peccati, Sole, Taqqu, and Utzet recently combined Stein's method andMalliavin calculus to obtain a bound for the Wasserstein distance of a Poissonfunctional and a Gaussian random variable. Convergence in the Wassersteindistance always implies convergence in the Kolmogorov distance at a possiblyweaker rate. But there are many examples of central limit theorems having thesame rate for both distances. The aim of this paper is to show this behaviourfor a large class of Poisson functionals, namely so-called U-statistics ofPoisson point processes. The technique used by Peccati et al. is modified toestablish a similar bound for the Kolmogorov distance of a Poisson functionaland a Gaussian random variable. This bound is evaluated for a U-statistic, andit is shown that the resulting expression is up to a constant the same as it isfor the Wasserstein distance.

Normal approximation of Poisson functionals in Kolmogorov distance