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 $k_n\in \{1,\dots ,n\}$ 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 $\Vert P_{E_n}{X}_n{\Vert }_2$ 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.

arXiv comment26 pages