Berry–Esseen bounds for random projections of $\ell _p^n$-balls
Volume 266 / 2022
Abstract
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 [D. Alonso-Gutiérrez et al., Bernoulli 25 (2019)]. 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-Gutiérrez et al. 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. We also discuss the rate of convergence in the central limit theorem of [Z. Kabluchko et al., Commun. Contemp. Math. {21} (2019)] for general $\ell _q$-norms of non-projected vectors chosen at random in an $\ell _p^n$-ball.
