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

Let X1, . . . , XN ∈ R be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has sup x∈Sn−1 ∣∣∣∣ 1 N N ∑ i=1 ( |〈Xi, x〉| − E|〈Xi, x〉| ) ∣∣∣∣ ≤ C√ n N , where C is an absolute positive constant. This result is valid in a more general framework when the linear forms (〈Xi, x〉)i≤N,x∈Sn−1 and the Euclidean norms (|Xi|/ √ n)i≤N exhibit uniformly a sub-exponential decay. As a consequence, if A denotes the random matrix with columns (Xi), then with overwhelming probability, the extremal singular values λmin and λmax of AA > satisfy the inequalities 1 − C √ n N ≤ λmin N ≤ λmax N ≤ 1 + C √ n N which is a quantitative version of Bai-Yin theorem [4] known for random matrices with i.i.d. entries.