In this paper, we prove a multivariate central limit theorem for ℓq-norms of highdimensional random vectors that are chosen uniformly at random in an ℓ n p -ball. As a consequence, we provide several applications on the intersections of ℓ n p -balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ℓ n p -ball onto a line spanned by a random direction θ ∈ S n−1 . The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime 1 ≤ p < q this displays in speed and rate function deviations of the q-norm on an ℓ n p -ball obtained by Schechtman and Zinn, but we obtain explicit constants.2010 Mathematics Subject Classification. 52A22, 60D05, 60F05, 60F10.

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In this paper, we study high-dimensional random projections of ℓ n p -balls. More precisely, for any n ∈ N let En be a random subspace of dimension kn ∈ {1, . . . , n} and Xn be a random point in the unit ball of ℓ n p . Our work provides a description of the Gaussian fluctuations of the Euclidean norm PE n Xn 2 of random orthogonal projections of Xn onto En. In particular, under the condition that kn → ∞ it is shown that these random variables satisfy a central limit theorem, as the space dimension n tends to infinity. Moreover, if kn → ∞ 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.2010 Mathematics Subject Classification. Primary: 52A22 Secondary: 46B07, 60F05.

Denote by λ 1 (A), . . . , λ n (A) the eigenvalues of an (n × n)-matrix A. Let Z n be an (n × n)-matrix chosen uniformly at random from the matrix analogue to the classical ℓ n pball, defined as the set of all self-adjoint (n × n)-matrices satisfying n k=1 |λ k (A)| p ≤ 1. We prove a large deviations principle for the (random) spectral measure of the matrix n 1/p Z n . As a consequence, we obtain that the spectral measure of n 1/p Z n converges weakly almost surely to a non-random limiting measure given by the Ullman distribution, as n → ∞. The corresponding results for random matrices in Schatten trace classes, where eigenvalues are replaced by the singular values, are also presented.

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