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

Abstract

In this paper, we prove three fundamental types of limit theorems for the [Formula: see text]-norm of random vectors chosen at random in an [Formula: see text]-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of [Formula: see text]-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Z. Kabluchko, J. Prochno and C. Thäle, High-dimensional limit theorems for random vectors in [Formula: see text]-balls, Commun. Contemp. Math. 21(1) (2019) 1750092].