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 ℓ n p -ball which has been obtained in [Alonso-Gutiérrez, Prochno, Thäle: Gaussian fluctuations for highdimensional random projections of ℓ n p -balls, Bernoulli 25(4A), 2019, 3139-3174]. More precisely, for any n ∈ N let En be a random subspace of dimension kn ∈ {1, . . . , n}, PE n the orthogonal projection onto En, and Xn be a random point in the unit ball of ℓ n p . We prove a Berry-Esseen theorem for PE n Xn 2 under the condition that kn → ∞. This answers in the affirmative a conjecture of Alonso-Gutiérrez, Prochno, and Thäle who obtained a rate of convergence under the additional condition that kn/n 2/3 → ∞ 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. In this work we also discuss the rate of convergence in the central limit theorem of [Kabluchko, Prochno, Thäle: High-dimensional limit theorems for random vectors in ℓ n p -balls, Commun. Contemp. Math. ( 2019)] for general ℓq-norms of non-projected vectors chosen at random in an ℓ n p -ball.

Large deviations, moderate deviations, and the KLS conjecture

“…This brings us to one of the main purposes of this paper, which is to establish a new connection between the KLS conjecture and the study of so-called large and moderate deviations principles for isotropic log-concave random vectors that provides a potential route to disprove the variance conjecture and thus the KLS conjecture. While the study of central limit theorems for random geometric quantities in asymptotic convex geometry is by now a classical and still flourishing part of the theory, as can be seen in [3,18,20,21,22,23,28,34,35], fluctuations beyond the Gaussian scale have only recently attracted attention in this context. For one, there are the large deviations, or more precisely large deviations principles, which determine the asymptotic likelihood of rare events on a scale of a law of large numbers, the most classical result in this direction being Cramér's theorem [10] (see also [12]).…”

“…= 0 while we know that they do satisfy the KLS conjecture (see [1,Theorem 2.3]). The crosspolytopes just discussed are members of the important class of ℓ n p -balls, whose probabilistic aspects have been under intensive investigation during the last years [2,3,15,16,20,22,23,27,33,37] (see also the comments in the introduction). As usual, for 1 ≤ p < ∞ we let B n p := {x ∈ R n : x p ≤ 1} denote the unit ball in R n with respect to the ℓ n p -norm…”

Thin-shell theory for rotationally invariant random simplices

“…As for the second bound, suppose p ≥ 7. Now rewriting (33) to obtain the first inequality below, and using the fact that p ≥ 7 to obtain the second, we have…”

“…Among others, there is Schmuckenschläger's central limit theorem related to the volume of intersections of n p -balls [52] and its multivariate version by Kabluchko, Prochno, and Thäle who also obtained moderate and large deviations principles [36,37]. Then there is the prominent central limit theorem for convex bodies proved by Klartag, showing that most lower-dimensional marginals of a random vector uniformly distributed in an isotropic convex body are approximately Gaussian [41], and a number of other results in which limit theorems related to analytic and geometric aspects of high-dimensional objects have been established [3,4,5,7,10,12,15,23,31,33,35,38,39,40,44,50,51,54,56,57].…”

“…Proof. For a proof of (81) see Lemma 2.5 and Proposition 2.6 of [33] The KS distance between two Gaussians with the same variance but different means may be bounded as follows.…”