Limit theorems for random points in a simplex

Abstract

In this work the ℓ q -norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry-Esseen bounds in the regime 1 ≤ q < ∞ are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where q = ∞. A comparison with corresponding results for ℓ n p -balls is carried out as well.

Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures

“…The past decades have shown the fundamental importance of central limit phenomena for both fields, the most prominent example being arguably the central limit theorem for convex bodies due to Klartag [39], which says that most marginals of an isotropic convex body in high dimensions are close to a Gaussian distribution. Beyond that, various geometric quantities have been shown to follow a central limit theorem as the dimension of the ambient space tends to infinity, e.g., [4,7,10,25,30,31,34,35,48,52,55,57,59], and aside from the universality they describe, which no doubt is a beautiful and fascinating property in its own right, those weak limit theorems find applications in different situations, e.g., [5,34,55]. What many of those results have in common and what makes their proofs more delicate is that the source of the Gaussian approximation is not attributed to independence, or a weak form of independence, but rather to geometry and more specifically convexity.…”

Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures

“…The past decades have shown the fundamental importance of central limit phenomena for both fields, the most prominent example being arguably the central limit theorem for convex bodies due to Klartag [38], which says that most marginals of an isotropic convex body in high dimensions are close to a Gaussian distribution. Beyond that, various geometric quantities have been shown to follow a central limit theorem as the dimension of the ambient space tends to infinity, e.g., [4,7,10,24,29,30,33,34,47,51,54,56,58], and aside from the universality they describe, which no doubt is a beautiful and fascinating property in its own right, those weak limit theorems find applications in different situations, e.g., [5,33,54]. What many of those results have in common and what makes their proofs more delicate is that the source of the Gaussian approximation is not attributed to independence, or a weak form of independence, but rather to geometry and more specifically convexity.…”

Berry–Esseen bounds for random projections of $\ell _p^n$-balls

“…Lemma 2.8. Let X (1) , X (2) , X (3) be (possibly dependent) random variables, and let G be a centered Gaussian random variable with variance σ 2 . Then, for any ε ∈ (0, ∞),…”

“…Other than Klartag's central limit theorem there are only a few random geometric quantities that have been shown to satisfy a central limit theorem. For instance, central limit phenomena for the log-volume of random simplices in high dimensions have recently been studied by Grote, Kabluchko, and Thäle [7], and a Berry-Esseen type central limit theorem for ℓ n q -norms of points chosen uniformly at random in a centered regular simplex in high dimensions was obtained by Baci, Kabluchko, Prochno, Sonnleitner, and Thäle [3]. Then there is the central limit theorem for the volume of k-dimensional random projections of the n-dimensional cube obtained by Paouris, Pivovarov, and Zinn [13], which in the case k = 1 was obtained independently by Kabluchko, Litvak, and Zaporozhets [8].…”