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 $\operatorname {Trace}f
\bigl(\Sigma_n\Sigma_n^*\bigr)=\sum_{i=1}^Nf(\lambda_i),\qquad (\lambda_i)\
eigenvalues\ of\ \Sigma_n\Sigma_n^*,$ 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 $\vert
\mathcal{V}\vert ^2=\bigl|\mathbb{E}\bigl(X_{11}^n\bigr) ^2\bigr|^2$ and
$\kappa=\mathbb{E}\bigl \vert X_{11}^n\bigr \vert ^4-\vert {\mathcal{V}}\vert
^2-2$ 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-Sj\"{o}strand's formula. The CLT is expressed in terms
of vanishing L\'{e}vy-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.Comment: Published at http://dx.doi.org/10.1214/15-AAP1135 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org