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

Peter Eichelsbacher (Ruhr University Bochum)
Christoph Thäle (Ruhr University Bochum)

Abstract


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-Lévy processes are considered.

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Pages: 1-25

Publication Date: October 28, 2014

DOI: 10.1214/EJP.v19-3061

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