parameter $\alpha$ indicates that most trajectories $X$ are such that $\eta(X)$ is far from $1/2$: this makes in a sense the classification problem easier.\
Our first contribution, detailed in Proposition \[thm:margin\] below, entails that the margin parameter associated with Model (\[eq:model\]) crucially depends on the distance between the functions $f$ and $g$ of interest. The proof is postponed to Appendix \[s:pmargin\].\

\[thm:margin\] Let $X$ be distributed according to Model (\[eq:model\]), and set $\Delta := \| f-g \|$. Then, for all $ 0 < {\varepsilon}{\leqslant}1/8$, we have $${\mathbb{P}}_{X} \left( \left| \eta(X) - \frac{1}{2} \right| {\leqslant}{\varepsilon}\right) {\leqslant}1 \wedge \frac{10 {\varepsilon}}{\Delta} \,.$$

In particular, if the distance $\| f-g \|$ is bounded from below by a positive constant, then  is satisfied with a margin parameter $\alpha=1$. If, instead, $\|f-g\|$ is allowed to be arbitrarily small, then nothing can be guaranteed about the margin parameter (except the obvious value $\alpha=0$ that always works).

Upper bounds on the excess risk {#s:upper}
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In this section we construct a classifier with nearly optimal excess risk. We detail its construction in Section \[sec:pluginFinitedim\] and analyze its approximation and estimation errors in Sections \[sec:trunc\] and \[sec:estimationerror\]. Our main result, Theorem \[prop:minmax\], is stated in Section \[subsection:upper\_bound\]. Nearly matching 