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

## TL;DR

This paper establishes central limit, moderate deviations, and large deviations theorems for the q-norms of high-dimensional random vectors in p^n-balls, extending previous work with new applications to projections.

## Contribution

It introduces a unified framework for limit theorems for p^n-ball vectors under general distributions, including new applications to projections.

## Key findings

- Proved a central limit theorem for p^n-ball vectors.
- Established moderate deviations principles.
- Derived large deviations results.

## Abstract

In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $\ell_p^n$-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, Gu\'edon, 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 $\ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Th\"ale: High-dimensional limit theorems for random vectors in $\ell_p^n$-balls, Commun. Contemp. Math. (2019)].