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

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

Gaussian fluctuations for edge counts in high-dimensional random geometric graphs

“…For a deeper discussion of Fock Spaces and Chaos Expansion as well as Malliavin-Calculus and Malliavin-Stein Method we refer the reader to [Las16] and the books [LP18; PR16]. We introduce the notion of the Wiener-Itô chaos expansion, see [LPS16] and the references therein, especially [LP11] for more details and proofs. Every Poisson functional F admits a representation of the type…”

“…This bound as well as the bound derived in [Pec12, Theorem 3.1], stated here as Theorem 3.1 for Poisson approximation in the total variation distance rely on the inverse L −1 of the Ornstein-Uhlenbeck generator L, which generally requires the calculation of the Wiener-Itô chaos expansion of F . In [LPS16] this was solved for the normal approximation case by establishing and applying a general Mehler formula for Poisson processes which allows to represent the inverse Ornstein-Uhlenbeck generator in terms of thinned Poisson point processes to derive bounds that only rely on the moments of the first-and second-order difference operators D x F and D 2 x 1 ,x 2 F .…”

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We prove a Poisson limit theorem in the total variation distance of functionals of a general Poisson point process using the Malliavin-Stein method. Our estimates only involve first and second order difference operators and are closely related to the corresponding bounds for the normal approximation in the Wasserstein distance by Last, Peccati and Schulte, see [LPS16]. As an application of this Poisson limit theorem, we consider a stationary Poisson point process in R d and connect any two points whenever their distance is less than or equal to a prescribed distance parameter.

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Stochastic Analysis for Poisson Point Processes

“…The reader will then enter the realm of the Stein and Chen-Stein methods for probabilistic approximations and be shown how to combine these techniques with Malliavin calculus operators (S. Bourguin [16]). The content of this chapter represents a substantial expansion and refinement of the seminal reference [14] and provides a self-contained account of several fundamental analytical results on the Poisson space (see e.g.…”

“…This chapter provides a detailed and unified discussion of a collection of recently introduced techniques (see e.g. [16,21,23,24]), allowing one to establish limit theorems for sequences of Poisson functionals with explicit rates of convergence, by combining Stein's method (see e.g. [5,19]) and Malliavin calculus.…”