Non-asymptotic bounds for Gaussian and bootstrap approximation have recently attracted significant interest in high-dimensional statistics. This paper studies Berry-Esseen bounds for such approximations with respect to the multivariate Kolmogorov distance, in the context of a sum of $n$ random vectors that are $p$-dimensional and i.i.d. Up to now, a growing line of work has established bounds with mild logarithmic dependence on $p$. However, the problem of developing corresponding bounds with near $n^{-1/2}$ dependence on $n$ has remained largely unresolved. Within the setting of random vectors that have sub-Gaussian or sub-exponential entries, this paper establishes bounds with near $n^{-1/2}$ dependence, for both Gaussian and bootstrap approximation. In addition, the proofs are considerably distinct from other recent approaches and make use of an "implicit smoothing" operation in the Lindeberg interpolation.
翻译:高山和靴杆近似的非保护界限最近引起了对高维统计的极大兴趣。 本文研究的是“ 格里- Esseen ” 的界限, 涉及多变科尔莫戈罗夫距离的近似界限, 其范围为约1美元随机矢量, 其价值为1美元和i. i. d. 。 迄今, 越来越多的工作线已经确立了对美元轻度对逻辑依赖的界限。 然而, 开发对美元的依赖程度接近 $ / / / / 美元的相应界限的问题基本上仍未解决。 在含有亚高山或亚爆炸性条目的随机矢量的设置中, 本文为高山和靴套近似近似值的矢量设定了近 $ / 1/ / / 美元的依赖性。 此外, 证据与最近的其他方法有很大区别, 并在林德贝格国际钻探中使用了“ 模糊的平滑动” 操作 。