Advances in Applied Probability

Moments and central limit theorems for some multivariate Poisson functionals

Günter Last, Mathew D. Penrose, Matthias Schulte, and Christoph Thäle

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This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in Rd.

Article information

Adv. in Appl. Probab., Volume 46, Number 2 (2014), 348-364.

First available in Project Euclid: 29 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes

Berry-Esseen-type bound central limit theorem intersection process multiple Wiener-Ito; integral Poisson process Poisson flat process product formula stochastic geometry Wiener-Ito chaos expansion


Last, Günter; Penrose, Mathew D.; Schulte, Matthias; Thäle, Christoph. Moments and central limit theorems for some multivariate Poisson functionals. Adv. in Appl. Probab. 46 (2014), no. 2, 348--364. doi:10.1239/aap/1401369698.

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