Title: Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles Show Chinese title

Title: 高维中心极限定理关于超矩形的维度依赖性注记

Abstract: Let $X_1,\dots,X_n$ be independent centered random vectors in $\mathbb{R}^d$. This paper shows that, even when $d$ may grow with $n$, the probability $P(n^{-1/2}\sum_{i=1}^nX_i\in A)$ can be approximated by its Gaussian analog uniformly in hyperrectangles $A$ in $\mathbb{R}^d$ as $n\to\infty$ under appropriate moment assumptions, as long as $(\log d)^5/n\to0$. This improves a result of Chernozhukov, Chetverikov & Kato [Ann. Probab. 45 (2017) 2309-2353] in terms of the dimension growth condition. When $n^{-1/2}\sum_{i=1}^nX_i$ has a common factor across the components, this condition can be further improved to $(\log d)^3/n\to0$. The corresponding bootstrap approximation results are also developed. These results serve as a theoretical foundation of simultaneous inference for high-dimensional models. Show Chinese abstract

Abstract: 设 $X_1,\dots,X_n$ 是独立的中心化随机向量在 $\mathbb{R}^d$。 本文表明，即使当 $d$ 可能随 $n$ 增长时，概率 $P(n^{-1/2}\sum_{i=1}^nX_i\in A)$ 在适当的矩假设下，只要 $(\log d)^5/n\to0$，就可以在 $\mathbb{R}^d$ 中的超矩形 $A$ 上一致地用其高斯近似来逼近，当 $n\to\infty$ 成立时。 这在维度增长条件方面改进了Chernozhukov、Chetverikov与Kato [Ann. Probab. 45 (2017) 2309-2353] 的结果。 当$n^{-1/2}\sum_{i=1}^nX_i$在组成成分间存在公因数时，这一条件可以进一步改进为$(\log d)^3/n\to0$。 相应的自助法（bootstrap）近似结果也被提出。 这些结果构成了高维模型同时推断的理论基础。