New Kolmogorov bounds for functionals of binomial point processes

TL;DR

This paper develops explicit Berry-Esseen bounds in the Kolmogorov distance for normal approximations of non-linear functionals of independent variables, extending previous Wasserstein bounds using Stein's method and difference operators.

Contribution

It introduces new Kolmogorov bounds for functionals of binomial point processes, generalizing prior Wasserstein bounds and providing new variance lower bounds via classical Hoeffding decompositions.

Findings

New Berry-Esseen bounds for set approximations with random tessellations

Bounds for functionals of covering processes

Enhanced variance lower bounds for non-linear functionals

Abstract

We obtain explicit Berry-Esseen bounds in the Kolmogorov distance for the normal approximation of non-linear functionals of vectors of independent random variables. Our results are based on the use of Stein's method and of random difference operators, and generalise the bounds recently obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry-Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of covering processes.