Electronic Journal of Probability

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

Peter Eichelsbacher and Christoph Thäle

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 102, 25 pp.

Accepted: 28 October 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G57: Random measures 60G55: Point processes
Secondary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G51: Processes with independent increments; Lévy processes

Berry-Esseen bound central limit theorem de Jong's theorem flat processes Malliavin calculus multiple stochastic integral Ornstein-Uhlenbeck-L\'evy process Poisson process random graphs random measure Stein's method U-statistics

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Eichelsbacher, Peter; Thäle, Christoph. New Berry-Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab. 19 (2014), paper no. 102, 25 pp. doi:10.1214/EJP.v19-3061. https://projecteuclid.org/euclid.ejp/1465065744

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