Berry-Esseen bounds for random projections of $\ell_p^n$-balls

Abstract

In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an $\ell_p^n$-ball which has been obtained in [Alonso-Guti\'errez, Prochno, Th\"ale: Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls, Bernoulli 25(4A), 2019, 3139--3174]. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$, $P_{E_n}$ the orthogonal projection onto $E_n$, and $X_n$ be a random point in the unit ball of $\ell_p^n$. We prove a Berry-Esseen theorem for $\|P_{E_n}X_n\|_2$ under the condition that $k_n\to\infty$. This answers in the affirmative a conjecture of Alonso-Guti\'errez, Prochno, and Th\"ale who obtained a rate of convergence under the additional condition that $k_n/n^{2/3}\to\infty$ as $n\to\infty$. In addition, we study the Gaussian fluctuations and Berry-Esseen bounds in a $3$-fold randomized setting where the dimension of the Grassmannian is also chosen randomly. Comparing deterministic and randomized subspace dimensions leads to a quite interesting observation regarding the central limit behavior. In this work we also discuss the rate of convergence in the central limit theorem of [Kabluchko, Prochno, Th\"ale: High-dimensional limit theorems for random vectors in $\ell_p^n$-balls, Commun. Contemp. Math. (2019)] for general $\ell_q$-norms of non-projected vectors chosen at random in an $\ell_p^n$-ball.