This is an announcement for the paper “High-dimensional limit theorems for random vectors in $\ell_p^n$-balls” by Zakhar Kabluchkohttps://arxiv.org/find/math/1/au:+Kabluchko_Z/0/1/0/all/0/1, Joscha Prochnohttps://arxiv.org/find/math/1/au:+Prochno_J/0/1/0/all/0/1, Christoph Thaelehttps://arxiv.org/find/math/1/au:+Thaele_C/0/1/0/all/0/1.
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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.09470