New Berry-Esseen bounds for non-linear functionals of Poisson random measures

By Peter Eichelsbacher, C. Thaele. 2013

56 citations recorded by the source corpus.

This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations on the Poisson space, we derive a bound, which is strictly smaller than what is available in the literature. This is applied to sequences of multiple integrals and sequences of Poisson functionals having a finite chaotic expansion. This leads to new Berry-Esseen bounds in a Poissonized version of de Jong's theorem for degenerate U-statistics. Moreover, geometric functionals of intersection processes of Poisson $k$-flats, random graph statistics of the Boolean model and non-linear functionals of Ornstein-Uhlenbeck-Levy processes are considered.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/04cfa79a918e32bcaa46e2d4b23272634cc6276a

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In Applied Mathematics

In Mathematical Physics

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Citations in the corpus

Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, background): “Our result for the normal approximation of U-statistics of Poisson point processes (see Theorem 4.2) is used by Reitzner, Schulte, and Th¨ale in [15] and [16] to derive central limit theorems with rates of convergence for the Kolmogorov distance for the total edge length of the Gilbert graph and for”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “2) our results provide substantial extensions of the findings contained in [16, 36, 39], which only considered linear and quadratic functionals.”
Normal approximation for stabilizing functionals (2017, result): “Ck(Ps,βs−1/γ) is a so-called Poisson U-statistic. Bounds for the normal approximation of such random variables were deduced, for example, in [34] and [21] for the Wasserstein distance and in [42] and [14] for the Kolmogorov distance. These results should also yield bounds similar to those in (5.26).”
Normal approximation and concentration of spectral projectors of sample covariance (2015, methodology): “New Berry–Esseen type bounds for U -statistics by Eichelsbacher and Thäle [6], based on Stein method and Malliavin calculus in a Poisson space, look very promising, but they have been proved only for a Poissonized version of U -statistics.”