Gaussian fluctuations for high-dimensional random projections of $\ell_{p}^{n}$-balls

Abstract

In this paper, we study high-dimensional random projections of $\ell_{p}^{n}$-balls. More precisely, for any $n\in\mathbb{N}$ let $E_{n}$ be a random subspace of dimension $k_{n}\in\{1,\ldots,n\}$ and $X_{n}$ be a random point in the unit ball of $\ell_{p}^{n}$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_{n}}X_{n}\|_{2}$ of random orthogonal projections of $X_{n}$ onto $E_{n}$. In particular, under the condition that $k_{n}\to\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_{n}\to\infty$ fast enough, we provide a Berry–Esseen bound on the rate of convergence in the central limit theorem. At the end, we provide a discussion of the large deviations counterpart to our central limit theorem.