Quantitative bounds for high-dimensional non-linear functionals of Gaussian processes

TL;DR

This paper provides explicit quantitative bounds on the convergence rates of high-dimensional non-linear Gaussian functionals under various distances, with novel sub-polynomial bounds under the hyper-rectangle distance.

Contribution

It introduces the first explicit sub-polynomial convergence bounds for high-dimensional Gaussian functionals under the hyper-rectangle distance, extending beyond i.i.d. cases.

Findings

Sub-polynomial convergence rate under $d_R$ distance.

Polynomial convergence under $d_{ ext{C}}$ and $d_W$ distances.

Explicit Berry-Esseen bounds for multiple statistical methods.

Abstract

In this paper, we establish explicit quantitative Berry-Esseen bounds in the hyper-rectangle distance $d_R$, the convex distance $d_{\mathscr{C}}$ and the $1$-Wasserstein distance $d_W$ for high-dimensional, non-linear functionals of Gaussian processes, allowing for strong dependence between variables. Our main result demonstrates that, under a smoothness assumption, the convergence rate under $d_R$ is sub-polynomial in the dimension and polynomial under $d_{\mathscr{C}}$ and $d_W$. To the best of our knowledge, our results under $d_R$ provide the first explicit sub-polynomial bound for high-dimensional, non-linear functionals of Gaussian processes beyond the i.i.d. setting. Building on this, we derive explicit Berry-Esseen bounds under both $d_R$ and $d_{\mathscr{C}}$ for multiple statistical examples, such as the method of moments, empirical characteristic functions, empirical moment-generating functions, and functional limit theorems in high-dimensional settings.