  We present an argument based on the multidimensional and the uniform central
limit theorems, proving that, under some geometrical assumptions between the
target function $T$ and the learning class $F$, the excess risk of the
empirical risk minimization algorithm is lower bounded by
\[\frac{\mathbb{E}\sup_{q\in Q}G_q}{\sqrt{n}}\delta,\] where $(G_q)_{q\in Q}$
is a canonical Gaussian process associated with $Q$ (a well chosen subset of
$F$) and $\delta$ is a parameter governing the oscillations of the empirical
excess risk function over a small ball in $F$.
