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# Strong Gaussian Approximation for the Sum of Random Vectors

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:1693-1715, 2022.

#### 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 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).