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

By Günter Last, Giovanni Peccati, Matthias Schulte. 2014

135 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/28763757ff0b340f0a5c82863ed4fbf4aa2ddfc3

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Normal approximation for stabilizing functionals (2017, methodology): “11) is the following marked version of a result from [22] (see Proposition 1.”
Lectures on the Poisson Process (2017, background): “3 was proved in [89] after Chatterjee [21] proved a corresponding result for Gaussian vectors.”
Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, unknown): “In [7], Last, Peccati, and Schulte further simplify the bounds for the normal approximation of Poisson functionals in Kolmogorov distance from Theorem 1.1 and [3] and apply them to several problems, where they provide new presumably optimal rates of convergence.”
Stochastic analysis for Poisson processes (2014, methodology): “In the present generality the result is in [16].”