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

Kabluchko, Zakhar; Prochno, Joscha; Thäle, Christoph

Authors

Zakhar Kabluchko

Joscha Prochno

Christoph Thäle



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
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
Contract Date Jun 28, 2018

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