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On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

A. PajorL. Pastur — 2009

Studia Mathematica

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix H ( 0 ) and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of H ( 0 ) and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

Random ε-nets and embeddings in N

Y. GordonA. E. LitvakA. PajorN. Tomczak-Jaegermann — 2007

Studia Mathematica

We show that, given an n-dimensional normed space X, a sequence of N = ( 8 / ε ) 2 n independent random vectors ( X i ) i = 1 N , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map Γ : N defined by Γ x = ( x , X i ) i = 1 N embeds X in N with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into N with asymptotically best possible relation between N, n, and ε.

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