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Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

Part of: Stochastic processes Limit theorems Geometric probability and stochastic geometry Stochastic analysis

Published online by Cambridge University Press:  22 February 2016

Günter Last ,
Mathew D. Penrose ,
Matthias Schulte  and
Christoph Thäle
Günter Last*
Affiliation:
Karlsruhe Institute of Technology
Mathew D. Penrose*
Affiliation:
University of Bath
Matthias Schulte*
Affiliation:
Karlsruhe Institute of Technology
Christoph Thäle*
Affiliation:
Ruhr University Bochum
*
∗ Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany.
∗∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: m.d.penrose@bath.ac.uk
∗ Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany.
∗∗∗∗∗ Postal address: Faculty of Mathematics, Ruhr University Bochum, 44801 Bochum, Germany. Email address: christoph.thaele@rub.de
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Abstract

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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 .

Keywords

Berry-Esseen-type boundcentral limit theoremintersection processmultiple Wiener—Itô integralPoisson processPoisson flat processproduct formulastochastic geometryWiener—Itô chaos expansion

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 348 - 364
Copyright
© Applied Probability Trust 

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