Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Consider a N ×n matrix Σn = 1 √ n R 1/2 n Xn, where Rn is a nonnegative definite Hermitian matrix and Xn is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues: Trace f(ΣnΣ ∗ n) = N ∑ i=1 f(λi), (λi) eigenvalues of ΣnΣ ∗ n, are shown to be gaussian, in the regime where both dimensions of matrix Σn go to infinity at the same pace and in the case where f is of class C, i.e. has three continuous derivatives. The main improvements with respect to Bai and Silverstein’s CLT [5] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is non-null, i.e. whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to |V| = |E(X 11) | and κ = E |X 11| 4 − |V| − 2 appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix Rn but also on its eigenvectors. Second, we relax the analyticity assumption over f by representing the linear statistics with the help of Helffer-Sjöstrand’s formula. The CLT is expressed in terms of vanishing Lévy-Prohorov distance between the linear statistics’ distribution and a Gaussian probability distribution, the mean and the variance of which depend upon N and n and may not converge. AMS 2000 subject classification: Primary 15A52, Secondary 15A18, 60F15.