Zakhar Kabluchko
High-dimensional limit theorems for random vectors in ℓpn-balls
Kabluchko, Zakhar; Prochno, Joscha; Thäle, Christoph
Authors
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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