Quantative bounds for high-dimensional non-linear functionals of Gaussian processes
Andreas Basse-O'Connor, David Kramer-Bang
TL;DR
The paper delivers explicit non-asymptotic Berry–Esseen bounds for high-dimensional, non-linear functionals of Gaussian processes in the hyper-rectangle distance $d_\mathcal{R}$, accommodating strong dependence. Central to the approach is a Malliavin–Stein framework that yields a main bound depending on networked factors: sample size $n$, dimension $d$, autocovariance structure $\rho$, regularity $(\beta,\kappa)$, and the minimal eigenvalue $\sigma_*^2$ of the correlation matrix. The results cover key statistical tools—method of moments, empirical characteristic functions, empirical moment generating functions, and finite-dimensional Breuer–Major convergence—and reveal sub-polynomial dimension dependence when $\beta>1/2$ (and mild conditions on $\sigma_*^2$); for $\beta=1/2$ the dependence can be polynomial in $d$. This advances Gaussian approximation theory for dependent, high-dimensional settings and provides practical, explicit rates for finite-sample inference across several statistical procedures. The methodology and bounds have significant implications for high-dimensional inference in time series and dependent data contexts where explicit dimension-aware Gaussian approximations are needed.
Abstract
In this paper, we derive explicit quantitative Berry-Esseen bounds in the hyper-rectangle distance for high-dimensional, non-linear functionals of Gaussian processes, allowing for strong dependence between variables. Our main result shows that the convergence rate is sub-polynomial in dimension $d$ under a smoothness assumption. Based on this result, we derive explicit Berry-Esseen bounds for the method of moments, empirical characteristic functions, empirical moment generation functions, and functional limit theorems in the high-dimensional setting.