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

Samuel G. G. Johnston, Joscha Prochno

Published 2019-11-02Version 1

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.