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

International audienceLetX1,...,XN ∈Rn,n≤N,beindependentcenteredrandomvectorswithlog-concavedistribution and with the identity as covariance matrix. We show that with overwhelming probability one has , decay. As a consequence, if A denotes the random matrix with columns (Xi), then with overwhelming probability, the extremﰅal singular values λmin and λmax of AA⊤ satisfy the inequalities 1 − Cﰅ n ≤ N λmin ≤ λmax ≤ 1 + C n which is a quantitative version of Bai-Yin theorem [4] known for random NNN matrices with i.i.d. entries