High-dimensional limit theorems for random vectors in $\ell_p^n$-balls

Abstract: 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.