Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles. (English) Zbl 1477.60048
Summary: Let \(X_1,\ldots ,X_n\) 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 \(n\rightarrow \infty\) under appropriate moment assumptions, as long as \((\log d)^5/n\rightarrow 0\). This improves a result of V. Chernozhukov et al. [Ann. Probab. 45, No. 4, 2309–2352 (2017; Zbl 1377.60040)] 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 \((\log d)^3/n\rightarrow 0\). The corresponding bootstrap approximation results are also developed. These results serve as a theoretical foundation of simultaneous inference for high-dimensional models.
MSC:
| 60F05 | Central limit and other weak theorems |
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
Keywords:
anti-concentration inequality; bootstrap; factor structure; maxima; randomized Lindeberg method; Stein kernelCitations:
Zbl 1377.60040References:
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