\
Note from  that the two subsamples $(X^0_i)_{1 {\leqslant}i {\leqslant}N_0}$ and $(X^1_i)_{1 {\leqslant}i {\leqslant}N_1}$ correspond to observations of the functions $g$ and $f$ respectively. Following our comments from Section \[sec:HilbertWhitenoise\], it is natural to define the random coefficients $(X_{i,j}^0)_{\substack{1 {\leqslant}i {\leqslant}N_0\\ 1 {\leqslant}j {\leqslant}d}}$ and $(X_{i,j}^1)_{\substack{1 {\leqslant}i {\leqslant}N_1\\ 1 {\leqslant}j {\leqslant}d}}$ by $$\label{eq:defX0X1}
\left\lbrace  
\begin{array}{l}
X_{i,j}^0 := \langle \varphi_j, X_i^0 \rangle  = \mu_j + {\varepsilon}_{i,j}^0 \,, \quad i=1\dots N_0 \,, \\
X_{i,j}^1 := \langle \varphi_j, X_i^1 \rangle = \theta_j + {\varepsilon}_{i,j}^1 \,, \quad i=1\dots N_1 \,,
\end{array}
\right.   \quad j\in \lbrace 1,\dots, d \rbrace ,$$ where the dimension $d \in \mathbb{N}^*$ will be determined later (as a function of $n$), and where $${\varepsilon}_{i,j}^k = \int_{0}^1 \varphi_{j}(t) dW_{\tau^k_i}(t) \,.$$ By independence of the random variables $Y_1,W_1,\ldots,Y_n,W_n$ used to generate the sample $(X_i,Y_i)_{1 {\leqslant}i {\leqslant}n}$ according to , and by the comments made in Section \[sec:HilbertWhitenoise\], we have the following conditional independence property for the ${\varepsilon}_{i,j}^k$.

\[rmk:conditionallyGaussian\] Conditionally on $(Y_1,\ldots,Y_n)$, the $n d$ random variables (or any $(Y_1,\ldots,Y_n)$-measurable permutation of them) $$\begin{array}{l}
{\varepsilon}^0_{1,1},\ldots,{\varepsilon}^0_{1,d},{\varepsilon}^0_{2,1},\ldots,{\varepsilon}^0_{2,d},\ldots,{\varepsilon}^0_{N_0,1},\ldots,{\varepsilon}^0_{N_0,d}, \\
{\varepsilon}^1_{1,1},\ldots,{\varepsilon}^1_{1,d},{\varepsilon}^1_{2,1},\ldots,{\varepsilon}^1_{2,d},\ldots,{\varepsilon}^1_{N_1,1},\ldots,{\varepsilon}^1_{N_1,d}
\end{array}$$ are i.i.d. ${\mathcal{N}}(0,1)$. As a consequence, on the event $\{N_0>0\} \cap \{N_1>0\}$, the random variables $N_k^{-1/2} \sum_{i=1}^{N_k} {\varepsilon}^k_{i,j}$, $1 {\leqslant}j {\leqslant}d$, $k \in \{0,1\}$, are i.i.d. 