$\Xi_n$ as $$\Xi_n=\left(\begin{array}{c}
\mathsf{E}_d\\
-\mathsf{E}_d
\end{array}\right)$$ in Theorem \[thm:main\], where $\mathsf{E}_d$ denotes the identity matrix of size $d$, we obtain the following result (note that continues to hold true while $\mathbb{R}^d$ is replaced by $(-\infty,\infty]^d$):

\[coro:main\] 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 $j$. Assume that the following convergences hold true: $$E\left[\|{\boldsymbol{X}}_n\|_{\ell_\infty}^2\|\Delta_n\|_{\ell_\infty}\right](\log d)^2\to0$$ and $$E\left[\left(1+\|{\boldsymbol{X}}_n\|_{\ell_\infty}^{|\nu|_*+1}\right)\left(1+\|Z_n\|_{\ell_\infty}^{|\nu_{\cdot 4}|}+\|\mathfrak{Z}_n\|_{\ell_\infty}^{|\nu_{\cdot 4}|}\right)\max_{1\leq j\leq d}\|\Delta_{n,j}(\nu)\|_{\ell_\infty}\right](\log d)^{\frac{3}{2}|\nu|_{**}+\frac{1}{2}}\to0$$ as $n\to\infty$ for every $\nu\in{\overline{\mathcal{N}}}_4^*({\overline{q}})$. Assume also that the following condition is satisfied: $$\lim_{b\downarrow0}\limsup_{n\to\infty}P(\min\operatorname{diag}(\mathfrak{C}_n)<b)=0.$$ Then we have $$\sup_{A\in\mathcal{A}^\mathrm{re}(d)}\left|P(Z_n\in A)-P(\mathfrak{Z}_n\in A)\right|\to0$$ as $n\to\infty$.

Some related results for statistical applications {#some-related-results-for-statistical-applications .unnumbered}
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In many applications, the objective variables are only approximately multiple Skorohod integrals. The following lemma is useful for such a situation.

\[lemma:approx\] For each $n\in\mathbb{N}$, let $Y_n,Y_n'$ be $m$-dimensional random vectors such that $$\sqrt{\log m}\|Y_n'-Y_n\|_{\ell_\infty}\to^p0$$ and $$\sup_{y\in\mathbb{R}^m}|P(Y_n\leq y)-P(\Xi_n\mathfrak{Z}_n\leq y)|\to0$$ as $n\to\infty$. Then we have $$\sup_{y\in\mathbb{R}^m}|P(Y'_n\leq y)-P(\Xi_n\mathfrak{Z}_n\leq y)|\to0$$ as $n\to\infty$, provided that holds true.

In terms of statistical applications, the mixed-normal approximation given by Theorem \[thm:main\] is often infeasible because the “asymptotic” covariance matrix $\mathfrak{C}_n$ usually contains unobservable quantities. In the following we give two auxiliary results bridging this gap. The first result ensures the validity of estimating the $\mathcal{F}$-conditional distribution of 