Concentration Property on Probability Spaces

Giannopoulos, Apostolos; Milman, Vitali

doi:10.1006/aima.2000.1949

Cited by 63 publications

(73 citation statements)

References 13 publications

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“…Theorem 3.1 is an empirical processes version of a lemma due to Bourgain ( [3], see also [4]) which deals with the case when F is S n−1 , considered as a class of linear functionals on R n and µ is an isotropic log-concave measure. Unlike Bourgain's argument, which relies heavily on the fact that the functions in the class are linear functionals and on that the indexing set is the whole sphere, Theorem 3.1 is very general.…”

Section: Decomposing Classes Of Functionsmentioning

confidence: 99%

“…Theorem 4.3 [21,4,6] For every 0 < ε, δ < 1 there is a constant c(ε, δ) for which the following holds. Let X 1 , ..., X k be independent random variables, distributed according to the volume measure of a convex, symmetric body in isotropic position.…”

Section: Sampling From An Isotropic Log-concave Measurementioning

confidence: 99%

“…Bourgain's proof uses a similar technique to the one we used here, which relies on the following version of Theorem 3.1. The formulation we present here is from [4]. Lemma 4.4 Let δ ∈ (0, 1) and let X 1 , ..., X k be points in R n sampled according to an isotropic log-concave measure.…”

Section: Sampling From An Isotropic Log-concave Measurementioning

confidence: 99%

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On weakly bounded empirical processes

Mendelson

2007

Math. Ann.

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Let F be a class of functions on a probability space (Ω, µ) and let X 1 , ..., X k be independent random variables distributed according to µ. We establish high probability tail estimates of the form sup f ∈F |{i : |f (X i )| ≥ t} using a natural parameter associated with F . We use this result to analyze weakly bounded empirical processes indexed by F and processes of the formWe also present some geometric applications of this approach, based on properties of the random operatorare sampled according to an isotropic, log-concave measure on R n .

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Section: Decomposing Classes Of Functionsmentioning

confidence: 99%

Section: Sampling From An Isotropic Log-concave Measurementioning

confidence: 99%

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On weakly bounded empirical processes

Mendelson

2007

Math. Ann.

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“…, m, uniformly distributed over a convex body in R n and the asymptotic geometry of the random convex polytope generated by these points (see e.g. [3,7,14,15,19]). …”

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confidence: 99%

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

Pajor

Pastur

2009

Studia Math.

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Abstract. We consider n × n real symmetric and hermitian random matrices Hn that are sums of a non-random matrix H (0) n 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) n 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.

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“…The question above was also investigated for any 0 < p < ∞. See, e.g., Bourgain [3], Rudelson [14], Giannopoulos and Milman [5], and Guédon and Rudelson [6]. We shall focus on the case of p = 1.…”

Section: Introductionmentioning

confidence: 99%