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Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

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

References

Bhattacharya, R. N. and Ghosh, J. K. (1992). A class of U-statistics and asymptotic normality of the number of k-clusters. J. Multivariate Anal. 43, 300–330.Google Scholar
Billingsley, P. (1979). “Probability and Measure.” John Wiley, New York.Google Scholar
Decreusefond, L., Ferraz, E., Randriambololona, H. and Vergne, A. (2014). Simplicial homology of random configurations. To appear in Adv. Appl. Prob. CrossRefGoogle Scholar
Heinrich, L. (2009). Central limit theorems for motion-invariant Poisson hyperplanes in expanding convex bodies. Rend. Circ. Mat. Palermo (2), Suppl. 81, 187–212.Google Scholar
Heinrich, L., Schmidt, H. and Schmidt, V. (2006). Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Prob. 16, 919–950.Google Scholar
Hug, D., Last, G. and Weil, W. (2004). A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272.Google Scholar
Kallenberg, O. (2002). “Foundations of Modern Probability”, 2nd edn. Springer, New York.Google Scholar
Lachièze-Rey, R. and Peccati, G. (2013). Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs. Electron. J. Prob. 18, 32.Google Scholar
Lachièze-Rey, R. and Peccati, G. (2013). Fine Gaussian fluctuations on the Poisson space, II: rescaled kernels, marked processes and geometric U-statistics. Stoch. Process. Appl. 123, 4186–4218.CrossRefGoogle Scholar
Last, G. and Penrose, M. D. (2011). Poisson process Fock space representation, chaos expansion and covariance inequalities. Prob. Theory Relat. Fields 150, 663–690.CrossRefGoogle Scholar
Minh, N. T. (2011). Malliavin–Stein method for multi-dimensional U-statistics of Poisson point processes. Preprint. Available at http://uk.arxiv.org/abs/1111.2140.Google Scholar
Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Springer, Milan.Google Scholar
Peccati, G. and Zheng, C. (2010). Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Prob. 15, 1487–1527.Google Scholar
Peccati, G., Solé, J. L., Taqqu, M. S. and Utzet, F. (2010). Stein's method and normal approximation of Poisson functionals. Ann. Prob. 38, 443–478.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.CrossRefGoogle Scholar
Reitzner, M. and Schulte, M. (2013). Central limit theorems for U-statistics of Poisson point processes. Ann. Prob. 41, 3879–3909.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Schulte, M. and Thäle, C. (2014). Distances between Poisson k-flats. Methodol. Comput. Appl. Prob. 16, 311–329.CrossRefGoogle Scholar
Surgailis, D. (1984). On multiple Poisson stochastic integrals and associated Markov semigroups. Prob. Math. Statist. 3, 217–239.Google Scholar