论文标题
高维度几乎最佳的中心极限定理和自举近似
Nearly optimal central limit theorem and bootstrap approximations in high dimensions
论文作者
论文摘要
在本文中，我们为高斯近似值得出了新的，几乎最佳的界限，以缩放平均$ n $独立的高维中心随机向量$ x_1，\ dots，x_n $在矩形的情况下，当缩放平均值的协方差矩阵中，平均值的矩形是非分数。在有限的$ x_i $的情况下，对kolmogorov的隐含限制在kolmogorov的平均分布和高斯向量之间的距离为$$ c（b^2_n \ log^3 d/n）^{1/2} {1/2} {1/2} \ log n，$$ n，$ d $是$ b_n $ $ b_ $ $ b_ $ b_ $ b_ $ b_ $ s的常数，$ b _ $ s n ins s s s compland s comportion s comportion s Suilt s Compents soncly s Compent x Unlance x Unlance x Unlance x Unlance x Unlance x Unbifect就$ d $和$ b_n $而言，这种界限是尖锐的，并且在样本量$ n $方面，这种界限几乎（最多可达$ \ log n $）。此外，我们表明乘数和经验引导近似值相似。此外，我们建立了允许无界$ x_i $的界限，仅根据$ x_i $的时刻来制定。最后，我们证明，在某些特殊的平滑和零稳定的情况下，可以进一步改善边界。
In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,\dots,X_n$ over the class of rectangles in the case when the covariance matrix of the scaled average is non-degenerate. In the case of bounded $X_i$'s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form $$C (B^2_n \log^3 d/n)^{1/2} \log n,$$ where $d$ is the dimension of the vectors and $B_n$ is a uniform envelope constant on components of $X_i$'s. This bound is sharp in terms of $d$ and $B_n$, and is nearly (up to $\log n$) sharp in terms of the sample size $n$. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded $X_i$'s, formulated solely in terms of moments of $X_i$'s. Finally, we demonstrate that the bounds can be further improved in some special smooth and zero-skewness cases.
