in $H$. Let $\theta$ be a cutoff function, that is a mapping $\theta:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}^+$ such that $\theta(x)=1$ if $\|x\|<1$, $\theta(x)=0$ for $\|x\|\geq 2$, and such that $\theta\in C_b^\infty({\mathbb{R}}^n)$. For $N\geq 1$, we set: $$G^N:=f^N(W(h_1),\cdots,W(h_n)), \quad f^N(x):=f(x) \times \theta(x/N), \; x \in {\mathbb{R}}^n.$$ Note that each random variable $G^N$ belongs to $\mathcal{S}$. Fix $r\geq 1$. We aim in proving that $\lim_{N\to +\infty} \|G^N-G\|_{1,r}=0$. On the one hand, $$\begin{aligned}
\E[|G^N-G|^r] &= \E\left[|G|^r {\left|\theta\left(\frac{W(h_1)}{N},\cdots,\frac{W(h_n)}{N}\right)-1\right|}^r\right]\\
&\leq \E[|G|^{2r}]^{1/2} \E\left[{\left|\theta\left(\frac{W(h_1)}{N},\cdots,\frac{W(h_n)}{N}\right)-1\right|}^{2r}\right]^{1/2}\\
&\leq C \int_{{\mathbb{R}}^n\setminus B^n(0,N)} e^{-\|x\|^2/2} dx,\end{aligned}$$ where $C$ is a positive constant. Hence, $\lim\limits_{N\to +\infty}\E[|G^N-G|^r] =0$. We now turn to the proof of the convergence of the derivatives. We have $$\nabla G^N = \sum_{i=1}^n \frac{\partial f^N}{\partial x_i}(W(h_1),\cdots,W(h_n)) h_i,$$ with $\frac{\partial f^N}{\partial x_i}(x) = \frac{\partial f}{\partial x_i} \theta(x/N) + N^{-1} f(x) \frac{\partial \theta}{\partial x_i}(x/N)$. Hence: $$\begin{aligned}
&\E\left[\|\nabla (G^N-G)\|_H^{2r}\right] = \sum_{i=1}^n \E\left[{\left|\frac{\partial f^N}{\partial x_i}-\frac{\partial f}{\partial x_i}\right|}^{2r}(W(h_1),\cdots,W(h_n))\right]\\
&\leq C \left(\sum_{i=1}^n \E\left[{\left|\frac{\partial f}{\partial x_i}(W(h_1),\cdots,W(h_n))\right|}^{2r} {\left|\theta\left(\frac{W(h_1)}{N},\cdots,\frac{W(h_n)}{N}\right)-1\right|}^{2r}\right]\right.\\
&\hspace{0.9em} \left. + N^{-2r}\E\left[{\left|(\frac{\partial \theta}{\partial x_i} \times f)(W(h_1),\cdots,W(h_n))\right|}^{2r} \right]\right)\\
&\underset{N\to +\infty}{\longrightarrow} 0.\end{aligned}$$

We conclude this section by introducing the following norms and spaces which are of interest when studying BSDEs. For any positive integers $p,n$, we set $\mathbb S_{n}^p$ the space of $\mathbb R^{ n}$-valued, continuous and $\mathbb F$-progressively measurable processes $Y$ s.t. $${\left\|Y\right\|}^p_{\mathbb 