AI 中文总结
本文证明了高维有界随机向量和的中心极限定理的最优上界，结合浓度估计与高斯比较方法，其误差界在维度增长时达到已知下界。
AI 中文摘要
设 $W=n^{-1/2}\sum_{i=1}^n X_i$，其中 $X_i$ 是 ${\mathbb R}^p$ 中的独立中心随机向量，且几乎必然满足 $|X_{ij}|\le B$。假设 $\text{Cov}(W)$ 具有单位对角元且最小特征值至少为 $b^2>0$。我们证明，$W$ 与具有相同协方差的高斯向量在轴对齐矩形上的一致距离至多为 $C\min\{1,b^{-2}Bn^{-1/2}\log^{3/2}(ep)\}$。对于固定的 $b$，对求和项大小和维度的依赖性与增长维度区域中的已知下界相匹配。证明结合了矩形边界附近的浓度估计与精心选择的高斯比较。
英文摘要
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most $C\min\{1,b^{-2}Bn^{-1/2}\log^{3/2}(ep)\}$. For fixed $b$, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.
