Vector-valued statistics of binomial processes: Berry-Esseen bounds in the convex distance

TL;DR

This paper establishes explicit bounds on the convex distance between the distribution of vector-valued functionals of i.i.d. random elements and Gaussian vectors, extending previous one-dimensional and smooth test function bounds to higher dimensions.

Contribution

It provides a novel multidimensional Berry-Esseen bound in the convex distance for vector-valued statistics, using Stein's method and recursive techniques, applicable to geometric functionals.

Findings

Explicit convex distance bounds in all dimensions

Rates of convergence with optimal sample size dependence

New multidimensional limit theorems for geometric functionals

Abstract

We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension. Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (2007) and Lachi\`eze-Rey and Peccati (2017), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (2019), and Fang and Koike (2021). Our techniques involve the use of Stein's method, combined with a suitable adaptation of the recursive approach inaugurated by Schulte and Yukich (2017): this yields rates of converge that have a presumably optimal dependence on the sample size. We develop several applications of a geometric nature, among which is a new collection of multidimensional quantitative limit theorems for the intrinsic volumes associated with coverage processes in Euclidean spaces.