Strong Gaussian Approximation for the Sum of Random Vectors

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 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 by distribution for the maximum norm in a high-dimensional setting ($p >n$).