how to establish an analogous result when the dimension $d$ possibly diverges as $n$ tends to infinity.

Indeed, even for the sum of independent random vectors, it is far from trivial to establish such a result in a situation where the dimension is possibly (much) larger than the sample size. This is not surprising because objective random vectors are typically not tight in the usual sense in such a high-dimensional setting, so any standard method to establish central limit theorems no longer works. A significant breakthrough in this subject was achieved by the seminal work of @CCK2013, where a Gaussian approximation of the maxima of the sum of independent random vectors in terms of the Kolmogorov distance has been established under quite mild assumptions which allow the dimension is (possibly exponentially) larger than the sample size. With the help of the Gaussian comparison theorem by @CCK2015, it enables us to construct feasible statistical inference procedures based on the maximum type statistics. Their theory, which we call the *Chernozhukov-Chetverikov-Kato theory*, or the *CCK theory* for short, has been developed in the subsequent work by @CCK2014 [@CCK2016] and @CCK2017: the first two papers have developed Gaussian approximation of the suprema of empirical 