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 ($p >n$).

翻译：本文为独立的随机矢量总和引出一个新的强大的高斯近似值。 这种方法依赖于最佳的运输理论,并产生对尺寸大小( $p$)和样本大小( $n$)的明显依赖。 这种依赖性为统计学习理论的所有实际应用设定了新的基本限制。 特别是,基于这一约束性,我们通过在高维环境中分配最高标准($p>n$)来证明近似值。