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Article Dans Une Revue Journal of the American Mathematical Society Année : 2010

Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

Radosław Adamczak
  • Fonction : Auteur
Alexander Litvak
  • Fonction : Auteur

Résumé

Let K be an isotropic convex body in Rn. Given ε > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to ε with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any ε > 0, there existsC(ε) > 0, suﰁch that if N ∼ C(ε)n and (Xi)i≤N are i.i.d. copies of ﱞﱞ1 N ﱞﱞ X, then ﱞN i=1 Xi ⊗ Xi − Idﱞ ≤ ε, with probability larger than 1 − exp(−c√n).
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Dates et versions

hal-00793769 , version 1 (22-02-2013)

Identifiants

  • HAL Id : hal-00793769 , version 1

Citer

Radosław Adamczak, Alexander Litvak, Alain Pajor, Nicole Tomczak-Jaegermann. Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles. Journal of the American Mathematical Society, 2010, 23, pp.535-561. ⟨hal-00793769⟩
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