论文标题
中央限制定理和bootstrap近似在高维度：接近$ 1/\ sqrt {n} $通过隐式平滑
Central Limit Theorem and Bootstrap Approximation in High Dimensions: Near $1/\sqrt{n}$ Rates via Implicit Smoothing
论文作者
论文摘要
高斯和自举近似的非反应界限最近引起了对高维统计的重大兴趣。本文在多元kolmogorov距离方面研究了浆果 - 埃斯尼的界限，以$ n $随机向量为$ p $ - 尺寸和i.i.d。到目前为止，越来越多的工作已经建立了对$ p $的轻度对数依赖性的界限。但是，开发与$ n^{ - 1/2} $依赖于$ n $的相应界限的问题基本上尚未解决。在具有亚高斯或次指定条目的随机向量的设置中，本文以接近$ n^{ - 1/2} $依赖关系建立边界，用于高斯和自举近似。此外，这些证明与其他最近的方法有很大不同，并利用了Lindeberg插值中的“隐式平滑”操作。
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.
