  In this paper, we prove a multivariate central limit theorem for
$\ell_q$-norms of high-dimensional random vectors that are chosen uniformly at
random in an $\ell_p^n$-ball. As a consequence, we provide several applications
on the intersections of $\ell_p^n$-balls in the flavor of Schechtman and
Schmuckenschl\"ager and obtain a central limit theorem for the length of a
projection of an $\ell_p^n$-ball onto a line spanned by a random direction
$\theta\in\mathbb S^{n-1}$. The latter generalizes results obtained for the
cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets.
Moreover, we complement our central limit theorems by providing a complete
description of the large deviation behavior, which covers fluctuations far
beyond the Gaussian scale. In the regime $1\leq p < q$ this displays in speed
and rate function deviations of the $q$-norm on an $\ell_p^n$-ball obtained by
Schechtman and Zinn, but we obtain explicit constants.
