This is an announcement for the paper “Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls” by David Alonso-Gutierrezhttps://arxiv.org/find/math/1/au:+Alonso_Gutierrez_D/0/1/0/all/0/1, Joscha Prochnohttps://arxiv.org/find/math/1/au:+Prochno_J/0/1/0/all/0/1, Christoph Thaelehttps://arxiv.org/find/math/1/au:+Thaele_C/0/1/0/all/0/1.
Abstract: In this paper, we study high-dimensional random projections of $\ell_p^n$-balls. More precisely, for any n∈ℕ let En be a random subspace of dimension $k_n\in{1,…,n}$ 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 $k_n\rightarrow\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_n\rightarrow\infty$ 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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.10130