Berry-Esseen bounds for weighted averages of Poisson avoidance functionals

TL;DR

This paper derives explicit Berry-Esseen bounds for the normal approximation of weighted averages of Poisson avoidance functionals using Malliavin calculus and Stein's method.

Contribution

It introduces a novel approach to obtain Berry-Esseen bounds for Poisson avoidance functionals via explicit Malliavin calculus expressions.

Findings

Bounds for the Wasserstein distance to normal distribution.

Application to volume of union of balls around Poisson points.

Bounds for avoidance functionals of empirical measures.

Abstract

We consider functionals which are weighted averages of the avoidance function of a Poisson process. Using the approach to Stein's method based on Malliavin calculus for Poisson functionals we provide explicit bounds for the Wasserstein distance between these standardized functionals and the standard normal distribution. Our approach relies on closed-form expressions for the action of some Malliavin type operators on avoidance functionals of Poisson processes. As a result we provide Berry-Esseen bounds in the CLT for the volume of the union of balls of a fixed radius around random Poisson centers or for the quantization error around points of a Poisson process. We also give Berry-Esseen bounds for avoidance functionals of empirical measures.