Strong Gaussian Approximation for the Sum of Random Vectors

TL;DR

This paper introduces a new strong Gaussian approximation bound for sums of independent random vectors, leveraging optimal transport theory to explicitly relate the approximation error to dimension and sample size, impacting high-dimensional statistical learning.

Contribution

It provides a novel Gaussian approximation bound with explicit dimension and sample size dependence, advancing understanding in high-dimensional probability and statistical learning.

Findings

Derived a new Gaussian approximation bound with explicit dependence on p and n

Established distributional approximation for the maximum norm in high-dimensional settings

Identified fundamental limits for applications in statistical learning

Abstract

This paper derives a new strong Gaussian approximation bound for the sum of independent random vectors. The approach relies on the optimal transport theory and yields \textit{explicit} dependence on the dimension size $p$ and the sample size $n$. This dependence establishes a new fundamental limit for all practical applications of statistical learning theory. Particularly, based on this bound, we prove approximation in distribution for the maximum norm in a high-dimensional setting ($p >n$).