${\mathcal{N}}(0,1)$ conditionally on $(Y_1,\ldots,Y_n)$.

 \
For every $j\in \lbrace 1, \dots, d \rbrace$, we use the coefficients defined in to estimate the coefficients $\mu_j$ and $\theta_j$ by $$\label{eq:defhatthethamu}
{\widehat}\mu_j = \mathds{1}_{\{N_0>0\}}\frac{1}{N_0} \sum_{i=1}^{N_0} X_{i,j}^0  \quad \mathrm{and} \quad {\widehat}\theta_j = \mathds{1}_{\{N_1>0\}} \frac{1}{N_1} \sum_{i=1}^{N_1} X_{i,j}^1 \,.$$ Note that we arbitrarily impose the value $0$ for ${\widehat}{\mu}_j$ when $N_0=0$ or for ${\widehat}{\theta}_j$ when $N_1=0$. This convention has a negligible impact, since with high probability $N_0$ and $N_1$ are both positive.

### A simple classifier {#sec:defclassifier}

We now build a simple classifier using the estimators ${\widehat}{\mu}_j$ and ${\widehat}{\theta}_j$ defined in . After observing a new trajectory $X = (X(t))_{0 {\leqslant}t {\leqslant}1}$, we construct the vector $\mathbf{X}_d \in {\mathbb{R}}^d$ defined by $$\mathbf{X}_d := \Bigl(\langle \varphi_1, X \rangle, \ldots, \langle \varphi_d, X \rangle\Bigr) \,.$$ Then, we assign the label $1$ to the trajectory $X$ if $\mathbf{X}_d$ is closer to ${\widehat}{\theta} := ({\widehat}\theta_1,\dots,{\widehat}\theta_d)$ than to ${\widehat}{\mu}:=({\widehat}\mu_1,\dots,{\widehat}\mu_d)$, and the label $0$ otherwise. More formally, our classifier ${\widehat}\Phi_d$ is defined for all trajectories $X$ by $${\widehat}\Phi_d(X) = \left\lbrace
\begin{array}{lcr}
1 & \mathrm{if} &  \| \mathbf{X}_d - {\widehat}\theta \|_d {\leqslant}\| \mathbf{X}_d- {\widehat}\mu \|_d  \\
0 & \mathrm{if} & \| \mathbf{X}_d - {\widehat}\theta \|_d > \| \mathbf{X}_d- {\widehat}\mu \|_d 
\end{array}
\right.  ,
