next result on the margin will be a key ingredient of our analysis.

Control of the margin in the functional model {#s:margin}
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As was shown in earlier works on binary supervised classification (see, [e.g.]{}, [@MT99] or [@AT07]), the probability mass of the region where the regression function $\eta$ is close to $1/2$ plays an important role in the convergence rates. The behaviour of the function $\eta$ is classically described by a so-called *margin assumption*: there exist $\alpha {\geqslant}0$ and ${\varepsilon}_0,C > 0$ such that, for all $0 < {\varepsilon}{\leqslant}{\varepsilon}_0$, $${\mathbb{P}}_{X} \left( \left| \eta(X) - \frac{1}{2} \right| {\leqslant}{\varepsilon}\right) {\leqslant}C {\varepsilon}^\alpha \,.
\label{eq:marginassumption}$$  \
We will show in Proposition \[thm:margin\] which parameters $\alpha,{\varepsilon}_0,C > 0$ are associated with Model (\[eq:model\]). The role of  is easy to understand: classifying a trajectory $X$ for which $\eta(X)$ is close to $1/2$ is necessarily a challenging problem because the events $\{Y=1\}$ and $\{Y=0\}$ are almost equally likely. This not only makes the optimal (Bayes) classifier $\mathds{1}_{\eta(X){\geqslant}1/2}$ error-prone, but it also makes the task of mimicking the Bayes classifier difficult. Indeed, any slightly bad approximation of $\eta$ when $\eta(X) \simeq 1/2$ can easily lead to a prediction different from $\mathds{1}_{\eta(X){\geqslant}1/2}$. A large value of the margin 