High-dimensional limit theorems for random vectors in ℓpn-balls

Abstract

In this paper, we prove a multivariate central limit theorem for [Formula: see text]-norms of high-dimensional random vectors that are chosen uniformly at random in an [Formula: see text]-ball. As a consequence, we provide several applications on the intersections of [Formula: see text]-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an [Formula: see text]-ball onto a line spanned by a random direction [Formula: see text]. 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 [Formula: see text] this displays in speed and rate function deviations of the [Formula: see text]-norm on an [Formula: see text]-ball obtained by Schechtman and Zinn, but we obtain explicit constants.