formally, let $\Xi_n$ be an $m\times d$ random matrix, where $m=m_n\geq{\textcolor{black}{3}}$ possibly depends on $n$. Our aim is to establish $$\label{aim}
\sup_{y\in\mathbb{R}^m}|P(\Xi_nZ_n\leq y)-P(\Xi_n\mathfrak{Z}_n\leq y)|\to0$$ as $n\to\infty$ under reasonable regularity conditions on $Z_n$ and $\Xi_n$. Mathematically speaking, given a vector $y\in\mathbb{R}^m$, the set $\{z\in\mathbb{R}^d:\Xi_nz\leq y\}$ is a finite intersection of hyperplanes in $\mathbb{R}^d$, i.e. convex polytopes in $\mathbb{R}^d$, the convergence can be considered as a high-dimensional central limit theorem for random convex polytopes. If we take $\Xi_n$ as the $d\times d$ diagonal matrix whose diagonals are the inverses of the “standard errors” of $Z_n$, the convergence does ensures the validity of (marginal) standardization of $Z_n$.

Now, our main theorem is stated as follows:

\[thm:main\] Suppose that $M_n,W_n\in\mathbb{D}_{\overline{q},\infty}(\mathbb{R}^d)$ and $\mathfrak{C}_n\in\mathbb{D}_{\overline{q},\infty}(\mathbb{R}^{d\times d})$ and that $u_n^j$ is symmetric for all $n$ and $j$. Suppose also that $\Xi_n$ can be written as $\Xi_n=\Upsilon_n\circ{\boldsymbol{X}}_n$ with $\Upsilon_n$ being an $m\times d$ (deterministic) matrix such that ${\@ifstar{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  s
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  }
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Upsilon_n}_\infty\geq1$ and ${\boldsymbol{X}}_{n}\in\mathbb{D}_{\overline{q},\infty}(\mathbb{R}^{m\times d})$. Assume that the following convergences hold true: $$\label{qtan-conv}
{\@ifstar{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  s
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  }
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Upsilon_n}_\infty^2E\left[\|{\boldsymbol{X}}_n\|_{\ell_\infty}^2\|\Delta_n\|_{\ell_\infty}\right](\log m)^2\to0$$ and $$\label{delta-conv}
{\@ifstar{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  s
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  }
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Upsilon_n}_\infty^{|\nu|_{**}+1}E\left[\left(1+\|{\boldsymbol{X}}_n\|_{\ell_\infty}^{|\nu|_*+1}\right)\left(1+\|Z_n\|_{\ell_\infty}^{|\nu_{\cdot 4}|}+\|\mathfrak{Z}_n\|_{\ell_\infty}^{|\nu_{\cdot 4}|}\right)\max_{1\leq j\leq d}\|\Delta_{n,j}(\nu)\|_{\ell_\infty}\right](\log m)^{\frac{3}{2}|\nu|_{**}+\frac{1}{2}}\to0$$ as $n\to\infty$ for every $\nu\in{\overline{\mathcal{N}}}_4^*({\overline{q}})$, where 