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November 2019 Gaussian fluctuations for high-dimensional random projections of $\ell_{p}^{n}$-balls
David Alonso-Gutiérrez, Joscha Prochno, Christoph Thäle
Bernoulli 25(4A): 3139-3174 (November 2019). DOI: 10.3150/18-BEJ1084


In this paper, we study high-dimensional random projections of $\ell_{p}^{n}$-balls. More precisely, for any $n\in\mathbb{N}$ let $E_{n}$ be a random subspace of dimension $k_{n}\in\{1,\ldots,n\}$ and $X_{n}$ be a random point in the unit ball of $\ell_{p}^{n}$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_{n}}X_{n}\|_{2}$ of random orthogonal projections of $X_{n}$ onto $E_{n}$. In particular, under the condition that $k_{n}\to\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_{n}\to\infty$ fast enough, we provide a Berry–Esseen bound on the rate of convergence in the central limit theorem. At the end, we provide a discussion of the large deviations counterpart to our central limit theorem.


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David Alonso-Gutiérrez. Joscha Prochno. Christoph Thäle. "Gaussian fluctuations for high-dimensional random projections of $\ell_{p}^{n}$-balls." Bernoulli 25 (4A) 3139 - 3174, November 2019.


Received: 1 October 2017; Revised: 1 August 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110124
MathSciNet: MR4003577
Digital Object Identifier: 10.3150/18-BEJ1084

Keywords: $\ell_{p}^{n}$-ball , Berry–Esseen bound , central limit theorem , cone measure , large deviations , random projection , uniform distribution

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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