  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.
