## Bernoulli

• Bernoulli
• Volume 25, Number 4A (2019), 3139-3174.

### Gaussian fluctuations for high-dimensional random projections of $\ell_{p}^{n}$-balls

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 3139-3174.

Dates
Revised: August 2018
First available in Project Euclid: 13 September 2019

https://projecteuclid.org/euclid.bj/1568362055

Digital Object Identifier
doi:10.3150/18-BEJ1084

Mathematical Reviews number (MathSciNet)
MR4003577

Zentralblatt MATH identifier
07110124

#### Citation

Alonso-Gutiérrez, David; Prochno, Joscha; Thäle, Christoph. Gaussian fluctuations for high-dimensional random projections of $\ell_{p}^{n}$-balls. Bernoulli 25 (2019), no. 4A, 3139--3174. doi:10.3150/18-BEJ1084. https://projecteuclid.org/euclid.bj/1568362055

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