非退化随机向量和的高维CLT：$n^{-1/2}$ -率
High-dimensional CLT for Sums of Non-degenerate Random Vectors: $n^{-1/2}$-rate
摘要：
当随机向量具有非奇异的协方差阵时，本文给出了高维矩形的Berry-Esseen界。在此非奇异性假设下，我们证明了一个$n^{-1/2}$标度的Berry-Esseen界的平均独立随机向量与有限的三阶矩的总和。该证明实质上是Senatov（2011）多元Berry-Esseen界的合成证明方法。与其他现有的工作（Kuchibhotla et al. 2018，Fang and Koike 2020 a）类似，本文考虑了高维情况下经典CLT证明技术的适用性和有效性。
英文摘要：
In this note, we provide a Berry--Esseen bounds for rectangles in high-dimensions when the random vectors have non-singular covariance matrices. Under this assumption of non-singularity, we prove an $n^{-1/2}$ scaling for the Berry--Esseen bound for sums of mean independent random vectors with a finite third moment. The proof is essentially the method of compositions proof of multivariate Berry--Esseen bound from Senatov (2011). Similar to other existing works (Kuchibhotla et al. 2018, Fang and Koike 2020a), this note considers the applicability and effectiveness of classical CLT proof techniques for the high-dimensional case.
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