Sharp bounds on the rate of convergence of the empirical covariance matrix

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• 刊名：Comptes Rendus Mathematique
• 出版年：2011
• 出版时间：February 2011
• 年：2011
• 卷：349
• 期：3-4
• 页码：195-200
• 全文大小：158 K

文摘

Let be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has where C is an absolute positive constant. This result is valid in a more general framework when the linear forms (X_i,x)_iN,xSⁿ⁻¹ and the Euclidean norms exhibit uniformly a sub-exponential decay. As a consequence, if A denotes the random matrix with columns (X_i), then with overwhelming probability, the extremal singular values λ_min and λ_max of AA satisfy the inequalities which is a quantitative version of Bai–Yin theorem (Z.D. Bai, Y.Q. Yin, 1993 [4]) known for random matrices with i.i.d. entries.