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

In this paper, we prove a multivariate central limit theorem for l(p)-norms of highdimensional random vectors that are chosen uniformly at random in an l(p)-ball. As a consequence, we provide several applications on the intersections of l(p)(n)-balls in the flavor of Schechtman and Schmuckenschlager and obtain a central limit theorem for the length of a projection of an l(p)(n)-ball onto a line spanned by a random direction theta is an element of Sn-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 ( )<= p < q this displays in speed and rate function deviations of the q-norm on an l(p)(n)-ball obtained by Schechtman and Zinn, but we obtain explicit constants.