2017/10/27 by David Alonso–Gutiérrez, Joscha Prochno, Alonso-Gutierrez, David +3
Mathematics · #46B07 #52A22 #60F05 #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1710.10130
openalex publication_date 2017/10/27 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In this paper, we study high-dimensional random projections of\n\ℓpn-balls. More precisely, for any n\∈ mathbb N let En be a random\nsubspace of dimension kn\∈ 1,\…,n and Xn be a random point in the\nunit ball of \ℓpn. Our work provides a description of the Gaussian\nfluctuations of the Euclidean norm \‖PEnXn\‖2 of random orthogonal\nprojections of Xn onto En. In particular, under the condition that\nkn\→\∞ it is shown that these random variables satisfy a central limit\ntheorem, as the space dimension n tends to infinity. Moreover, if\nkn\→\∞ fast enough, we provide a Berry-Esseen bound on the rate of\nconvergence in the central limit theorem. At the end we provide a discussion of\nthe large deviations counterpart to our central limit theorem.\n