GAUSSIAN FLUCTUATIONS FOR LINEAR SPECTRAL STATISTICS OF LARGE RANDOM COVARIANCE MATRICES

Consider a N x n matrix Sigma(n) = 1 root n R-n(1/2) X-n, where R-n is a nonnegative definite Hermitian matrix and X-n is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues Trace f (Sigma(n) Sigma(n)*) = Sigma(N)(i=1) f (lambda(i)), (lambda(i)) eigenvalues of Sigma(n) Sigma(n)*, are shown to be Gaussian, in the regime where both dimensions of matrix Sigma(n) go to infinity at the same pace and in the case where f is of class C-3, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein's CLT [Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to vertical bar V vertical bar(2) = vertical bar E(X-11(n))(2)vertical bar(2) and kappa = E vertical bar X-11(n)vertical bar(4) - vertical bar V vertical bar(2) - 2 appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix R-n but also on its eigenvectors. Second, we relax the analyticity assumption over f by representing the linear statistics with the help of Helffer-Sjostrand's formula. The CLT is expressed in terms of vanishing Levy-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.