$$\label{qtan}
\Delta_n=\left(\langle D^{q_j}M^i_n,u_n^j\rangle_{H^{\otimes q_j}}-\mathfrak{C}_n^{ij}\right)_{1\leq i,j\leq d}$$ and $$\label{eq:delta}
\Delta_{n,j}(\nu):=\left\langle \bigotimes_{k=1}^{q_j}(D^kM_n)^{\otimes \nu_{k1}}\otimes(D^k\mathfrak{C}_n)^{\otimes \nu_{k2}}\otimes(D^kW_n)^{\otimes \nu_{k3}}\otimes(D^k{\boldsymbol{X}}_{n})^{\otimes \nu_{k4}},u_n^j\right\rangle_{H^{\otimes q_j}}$$ Assume also that the following condition is satisfied: $$\label{diag-tight}
\lim_{b\downarrow0}\limsup_{n\to\infty}P(\min\operatorname{diag}(\Xi_n\mathfrak{C}_n\Xi_n^\top)<b)=0.$$ Then we have as $n\to\infty$.

The variable $\Delta_n$ defined in is called the *quasi-tangent* in [@NY2017].

The variable $\Delta_{n,j}(\nu)$ defined in takes values in $$\mathbb{R}^{\underbrace{\scriptstyle{d\times\cdots\times d}}_{|\nu|_{*}}\times\underbrace{\scriptstyle{m\times\cdots\times m}}_{|\nu_{\cdot4}|}
\times\underbrace{\scriptstyle{d\times\cdots\times d}}_{|\nu_{\cdot4}|}}$$ when $\nu=(\nu_{kl})\in\bigcup_{\alpha\in\mathcal{A}(q_j)}\mathcal{N}_4^*(\alpha)$. To see this, let us recall that $D^kM_n$, $D^k\mathfrak{C}_n$, $D^kW_n$ and $D^k{\boldsymbol{X}}_n$ take values in $(H^{\otimes k})^d$, $(H^{\otimes k})^{d\times d}$, $(H^{\otimes k})^d$ and $(H^{\otimes k})^{m\times d}$, respectively (cf. Section \[sec:malliavin\]). Therefore, according to the notation defined by , the variable $$\bigotimes_{k=1}^{q_j}(D^kM_n)^{\otimes \nu_{k1}}\otimes(D^k\mathfrak{C}_n)^{\otimes \nu_{k2}}\otimes(D^kW_n)^{\otimes \nu_{k3}}\otimes(D^k{\boldsymbol{X}}_{n})^{\otimes \nu_{k4}}$$ takes values in $$\left(H^{\otimes \sum_{k=1}^{q_j}k(\nu_{k1}+\nu_{k2}+\nu_{k3}+\nu_{k4})}\right)^{\underbrace{\scriptstyle{d\times\cdots\times d}}_{\sum_{k=1}^{q_j}(\nu_{k1}+2\nu_{k2}+\nu_{k3})}
\times\underbrace{\scriptstyle{m\times\cdots\times m}}_{\sum_{k=1}^{q_j}\nu_{k4}}
\times\underbrace{\scriptstyle{d\times\cdots\times d}}_{\sum_{k=1}^{q_j}\nu_{k4}}}
=\left(H^{\otimes q_j}\right)^{\underbrace{\scriptstyle{d\times\cdots\times d}}_{|\nu|_{*}}\times\underbrace{\scriptstyle{m\times\cdots\times m}}_{|\nu_{\cdot4}|}
\times\underbrace{\scriptstyle{d\times\cdots\times d}}_{|\nu_{\cdot4}|}},$$ where the last identity follows from the relation $\sum_{k=1}^{q_j}k(\nu_{k1}+\nu_{k2}+\nu_{k3}+\nu_{k4})=q_j$. Hence, according to the notation defined by , we obtain $$\begin{aligned}
\Delta_{n,j}(\nu)&=\left\langle \bigotimes_{k=1}^{q_j}(D^kM_n)^{\otimes \nu_{k1}}\otimes(D^k\mathfrak{C}_n)^{\otimes \nu_{k2}}\otimes(D^kW_n)^{\otimes \nu_{k3}}\otimes(D^k{\boldsymbol{X}}_{n})^{\otimes \nu_{k4}},u_n^j\right\rangle_{H^{\otimes q_j}}\\
&\in\mathbb{R}^{\underbrace{\scriptstyle{d\times\cdots\times d}}_{|\nu|_{*}}\times\underbrace{\scriptstyle{m\times\cdots\times m}}_{|\nu_{\cdot4}|}
\times\underbrace{\scriptstyle{d\times\cdots\times d}}_{|\nu_{\cdot4}|}}.\end{aligned}$$

In Theorem \[thm:main\], we require all the variables appearing there to have finite moments of all orders just for simplicity. It would be enough for them to have finite moments up to order $p$ only, where $p$ would be a function of ${\overline{q}}$.

As in the original CCK theory, it is 