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1
Content available remote

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

100%
Pajor A., Pastur L.
Studia Mathematica
|
2009
|
tom 195
|
nr 1
11-29
EN
We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix $Hₙ^{(0)}$ 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 $Hₙ^{(0)}$ and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges weakly in probability to the non-random limit, found by Marchenko and Pastur.
2
Content available remote

Regularization of star bodies by random hyperplane cut off

100%
Milman V., Pajor A.
Studia Mathematica
|
2003
|
tom 159
|
nr 2
247-261
EN
We present a general result on regularization of an arbitrary convex body (and more generally a star body), which gives and extends global forms of a number of well known local facts, like the low M*-estimates, large Euclidean sections of finite volume-ratio spaces and others.
3
Content available remote

Inequalities of the Kahane-Khinchin type and sections of $L_{p}$-balls

81%
Koldobsky A., Pajor A., Yaskin V.
Studia Mathematica
|
2008
|
tom 184
|
nr 3
217-231
EN
We extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in $L_{p}$, p > -2.
4
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Random ε-nets and embeddings in $ℓ^{N}_{∞}$

81%
Gordon Y., Litvak A., Pajor A., Tomczak-Jaegermann N.
Studia Mathematica
|
2007
|
tom 178
|
nr 1
91-98
EN
We show that, given an n-dimensional normed space X, a sequence of $N = (8/ε)^{2n}$ 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 $Γ: ℝ → ℝ^{N}$ defined by $Γx = (⟨x,X_{i}⟩)_{i=1}^{N}$ embeds X in $ℓ^{N}_{∞}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{∞}^{N}$ with asymptotically best possible relation between N, n, and ε.
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