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Gaussian fluctuations for high-dimensional random projections of \(\ell_p^n\)-balls. (English) Zbl 1431.60011

Summary: 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.

MSC:

60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
60F10 Large deviations
60G15 Gaussian processes
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References:

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