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Probability Theory and Related Fields

, Volume 165, Issue 3–4, pp 667–723 | Cite as

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization

  • Günter Last
  • Giovanni Peccati
  • Matthias Schulte
Article

Abstract

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincaré inequality, as well as on the use of Malliavin operators, of Stein’s method, and of an (integrated) Mehler’s formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals.

Keywords

Central limit theorem Chaos expansion Kolmogorov distance Malliavin calculus Mehler’s formula Nearest neighbour graph Poincaré inequality Poisson process Spatial Ornstein-Uhlenbeck process Stabilization Stein’s method Stochastic geometry Voronoi tessellation Wasserstein distance 

Mathematics Subject Classification

60F05 60H07 60G55 60D05 60G60 

Notes

Acknowledgments

GP wishes to thank Raphaël Lachièze-Rey for useful discussions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Günter Last
    • 1
  • Giovanni Peccati
    • 2
  • Matthias Schulte
    • 1
  1. 1.Institute of StochasticsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Mathematics Research UnitLuxembourg University LuxembourgLuxembourg

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