In the literature of high-dimensional central limit theorems, there is a gap between results for general limiting correlation matrix $\Sigma$ and the strongly non-degenerate case. For the general case where $\Sigma$ may be degenerate, under certain light-tail conditions, when approximating a normalized sum of $n$ independent random vectors by the Gaussian distribution $N(0,\Sigma)$ in multivariate Kolmogorov distance, the best-known error rate has been $O(n^{-1/4})$, subject to logarithmic factors of the dimension. For the strongly non-degenerate case, that is, when the minimum eigenvalue of $\Sigma$ is bounded away from 0, the error rate can be improved to $O(n^{-1/2})$ up to a $\log n$ factor. In this paper, we show that the $O(n^{-1/2})$ rate up to a $\log n$ factor can still be achieved in the degenerate case, provided that the minimum eigenvalue of the limiting correlation matrix of any three components is bounded away from 0. We prove our main results using Stein's method in conjunction with previously unexplored inequalities for the integral of the first three derivatives of the standard Gaussian density over convex polytopes. These inequalities were previously known only for hyperrectangles. Our proof demonstrates the connection between the three-components condition and the third moment Berry--Esseen bound.

翻译：在高维中心极限定理的文献中，关于一般极限相关矩阵$\Sigma$的结果与强非退化情形之间存在差距。对于$\Sigma$可能退化的广义情形，在特定轻尾条件下，用高斯分布$N(0,\Sigma)$近似$n$个独立随机向量归一化和的多变量Kolmogorov距离时，已知最优误差率为$O(n^{-1/4})$，且受限于维度的对数因子。对于强非退化情形（即$\Sigma$的最小特征值远离0），误差率可改进至$O(n^{-1/2})$（最多包含$\log n$因子）。本文证明，若任意三个分量的极限相关矩阵的最小特征值均远离0，则在退化情形下仍可达到$O(n^{-1/2})$的误差率（最多包含$\log n$因子）。我们通过Stein方法结合凸多面体上标准高斯密度前三阶导数积分的前沿不等式证明主要结论。这些不等式此前仅对超矩形成立。本文证明揭示了三分量条件与第三矩Berry-Esseen界之间的关联。