denote by $\mathbb{Z}_+$ the set of all non-negative integers. We define $$\mathcal{A}(q):=\{\alpha\in\mathbb{Z}_+^q:\alpha_1+2\alpha_2+\cdots+q\alpha_q=q\}.$$ For a multi-index $\alpha=(\alpha_1,\dots,\alpha_q)\in\mathbb{Z}_+^q$, we set $|\alpha|=\alpha_1+\cdots+\alpha_q$ as usual. Given another positive integer $r$, we define $$\mathcal{N}_r(\alpha):=\left\{\nu=(\nu_{ij})_{(i,j)\in[q]\times[r]}:\nu_{ij}\in\mathbb{Z},\sum_{j=1}^r\nu_{ij}=\alpha_i \right\}$$ and $$\mathcal{N}^*_r(\alpha)=\{\nu=(\nu_{ij})\in\mathcal{N}_r(\alpha):\nu_{q1}=0\}.$$ Moreover, we define $${\textcolor{black}{{\overline{\mathcal{A}}}(q):=\bigcup_{p=1}^q\mathcal{A}(p)\quad\text{and}\quad}}
{\overline{\mathcal{N}}}^*_r(q):=\bigcup_{\alpha\in{\textcolor{black}{{\overline{\mathcal{A}}}}}(q)}\mathcal{N}^*_r(\alpha).$$ Finally, for an element $\nu=(\nu_{ij})\in\mathcal{N}_4(\alpha)$, we set $|\nu|_*:=|\nu_{\cdot 1}|+2|\nu_{\cdot 2}|+|\nu_{\cdot 3}|$ and $|\nu|_{**}:=|\nu|_*+|\nu_{\cdot4}|$.

Main results {#sec:main}
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Throughout the paper, we consider an asymptotic theory such that the parameter $n\in\mathbb{N}$ tends to infinity. For each $n\in\mathbb{N}$, we consider a probability space $(\Omega^{n},\mathcal{F}^{n},P^{n})$, and we suppose that all the random variables at stage $n$ are defined on $(\Omega^{n},\mathcal{F}^{n},P^{n})$. We also suppose that an isonormal Gaussian process $\mathbb{W}_n=(\mathbb{W}_n(h))_{h\in H_n}$ over a real separable Hilbert space $H_n$ is defined on $(\Omega^{n},\mathcal{F}^{n},P^{n})$. To keep the notation simple, we subtract the indices $n$ from $(\Omega^{n},\mathcal{F}^{n},P^{n})$, $\mathbb{W}_n$ and $H_n$, respectively. So we will write them simply as $(\Omega,\mathcal{F},P)$, $\mathbb{W}$ and $H$, respectively. In particular, note that the spaces and the operators associated with $\mathbb{W}$ (which are introduced in Section \[sec:malliavin\]) implicitly depend on $n$, although we do not attach the index $n$ to them.

For each $n\in\mathbb{N}$, let $M_n$ be a $d$-dimensional random vector consisting of multiple Skorohod integrals: $$M_n^j=\delta^{q_j}(u_n^j),\qquad j=1,\dots,d,$$ where $q_j$ is a positive integer and $u_n^j\in\operatorname{Dom}(\delta^{q_j})$ 