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

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 Chernozhukov et al. (Ann Probab 45:2309–2353, 2017) 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.