High-dimensional limit theorems for random vectors in ℓpn-balls

Kabluchko, Zakhar; Prochno, Joscha; Thäle, Christoph

Abstract

In this paper, we prove a multivariate central limit theorem for ℓq-norms of high-dimensional random vectors that are chosen uniformly at random in an ℓnp-ball. As a consequence, we provide several applications on the intersections of ℓnp-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ℓnp-ball onto a line spanned by a random direction θ∈Sn−1. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime 1 is equal or smaller than p is smaller than q this displays in speed and rate function deviations of the q-norm on an ℓnp-ball obtained by Schechtman and Zinn, but we obtain explicit constants.

Citation

Kabluchko, Z., Prochno, J., & Thäle, C. (2019). High-dimensional limit theorems for random vectors in ℓpn-balls. Communications in contemporary mathematics, 21(1), 1750092. https://doi.org/10.1142/S0219199717500924

• Journal Article Type: Article
• Acceptance Date: Oct 9, 2017
• Online Publication Date: Nov 13, 2017
• Publication Date: 2019
• Deposit Date: Jun 28, 2018
• Publicly Available Date: Jul 3, 2018
• Journal: Communications in Contemporary Mathematics
• Print ISSN: 0219-1997
• Electronic ISSN: 1793-6683
• Publisher: World Scientific Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 1
• Article Number: 1750092
• Pages: 1750092
• DOI: https://doi.org/10.1142/S0219199717500924
• Keywords: Asymptotic geometric analysis; Central limit theorem; Extreme value distribution; High dimensions; Large deviation principle; ℓp-ball; Multivariate central limit theorem; Non-central limit theorem
• Public URL: https://hull-repository.worktribe.com/output/898849
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S0219199717500924
• Related Public URLs: https://arxiv.org/abs/1709.09470