Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles

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 $P(n^{-1/2}{\sum }_{i=1}^n{X}_i\in A)$ 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 $(\log d{)}^5/n\to 0$. This improves a result of Chernozhukov, Chetverikov & Kato [Ann. Probab. 45 (2017) 2309-2353] in terms of the dimension growth condition. When $n^{-1/2}{\sum }_{i=1}^n{X}_i$ has a common factor across the components, this condition can be further improved to $(\log d{)}^3/n\to 0$. The corresponding bootstrap approximation results are also developed. These results serve as a theoretical foundation of simultaneous inference for high-dimensional models.

arXiv comment33 pages. The constant of Lemma 2.2 is modified and the proof is corrected

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• Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
• Gaussian approximation of suprema of empirical processes
• Uniform Post Selection Inference for LAD Regression and Other Z-estimation problems
• Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications
• Stein kernels and moment maps
• Stein's method for normal approximation in Wasserstein distances with application to the multivariate Central Limit Theorem
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• Bootstrapping ${\ell }_p$-Statistics in High Dimensions