setting, this issue is usually resolved by proving the stability of the convergence so that $$(Z_n,X)\to^{\mathcal{L}}(\mathfrak{Z}_n,X)\qquad\text{as }n\to\infty$$ for any $m$-dimensional ($\mathcal{F}$-measurable) random variable $X$, where $\to^{\mathcal{L}}$ denotes the convergence in law. This statement is no longer meaningful in a high-dimensional setting such that $d\to\infty$ as $n\to\infty$, so we need to reformulate it appropriately. A naïve idea is to consider the following statement: $$\label{stable-kol}
\sup_{z\in\mathbb{R}^d,x\in\mathbb{R}^m}\left|P(Z_n\leq z,X\leq x)-P(\mathfrak{Z}_n\leq z,X\leq x)\right|\to0\qquad\text{as }n\to\infty.$$ However, if $m$ depends also on $n$, this type of statement is not attractive neither theoretical nor practical points of view due to the following reasons: From a theoretical point of view, we need to assume a so-called *anti-concentration inequality* for $X$ to prove this type of result by the CCK approach, but it is usually hard to check such an inequality for general random variables, especially when $m\to\infty$ as $n\to\infty$. Besides, from a practical point of view, it is still unclear whether the convergence ensures the validity of standardization of $Z_n$ because no analog of the continuous mapping theorem has been established yet for high-dimensional central limit theorems of the form . For these reasons we choose the way to directly prove convergence results for normalized statistics of $Z_n$. More 