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

Last, Günter; Peccati, Giovanni; Schulte, Matthias (2016). Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization. Probability theory and related fields, 165(3), pp. 667-723. Springer 10.1007/s00440-015-0643-7

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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.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science

UniBE Contributor:

Schulte, Matthias

Subjects:

500 Science > 510 Mathematics

ISSN:

0178-8051

Publisher:

Springer

Language:

English

Submitter:

David Ginsbourger

Date Deposited:

25 Apr 2017 16:59

Last Modified:

25 Apr 2017 17:00

Publisher DOI:

10.1007/s00440-015-0643-7

BORIS DOI:

10.7892/boris.93223

URI:

https://boris.unibe.ch/id/eprint/93223

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