Berry-Esseen bounds for random projections of Lp-Balls

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 ℓnp-ball which has been obtained in [D. Alonso-Gutiérrez et al., Bernoulli 25 (2019)]. More precisely, for any n∈N let En be a random subspace of dimension kn∈{1,…,n}, PEn the orthogonal projection onto En, and Xn be a random point in the unit ball of ℓnp. We prove a Berry–Esseen theorem for ∥PEnXn∥2 under the condition that kn→∞. This answers in the affirmative a conjecture of Alonso-Gutiérrez et al. who obtained a rate of convergence under the additional condition that kn/n2/3→∞ as n→∞. 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 ℓq-norms of non-projected vectors chosen at random in an ℓnp-ball.