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

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.