Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Abstract

Consider a $N\times n$ matrix $\Sigma_{n}=\frac{1}{\sqrt{n}}R_{n}^{1/2}X_{n}$, 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

\[\operatorname{Trace}f(\Sigma_{n}\Sigma_{n}^{*})=\sum_{i=1}^{N}f(\lambda_{i}),\qquad (\lambda_{i})\mbox{ eigenvalues of }\Sigma_{n}\Sigma_{n}^{*},\] 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

\[|\mathcal{V}|^{2}=|\mathbb{E}(X_{11}^{n})^{2}|^{2}\quad\mbox{and}\quad\kappa=\mathbb{E}\vert X_{11}^{n}\vert^{4}-\vert\mathcal{V} \vert^{2}-2\] 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–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.