I am reading Peter Hall's "the bootstrap and edgeworth expansion". In Theorem 2.3 on page 57, it claims that if the characteristic function $\chi$ of a $d$-dimension random variable $\mathbf X$ satisfy
$$ \limsup_{\|t\| \to \infty}|\chi(t)|<1, $$
then for the sample mean $\overline{\mathbf X}$ one has for some $\epsilon>0$,
$$ \sup_{x \in \mathbb R^d} \mathbb P(\overline{\mathbf X} = \mathbf x) = O(e^{-\epsilon n}), $$
as $n\to\infty$. I do not understand what $\mathbb P(\overline{\mathbf X} = \mathbf x)$ means here. Density? Probability? Isn't it always $0$ for a continuous random variable. Also what does this result imply, please? Thank you.