  We consider large-dimensional Hermitian or symmetric random matrices of the
form $W=M+\vartheta V$ where $M$ is a Wigner matrix and $V$ is a real diagonal
matrix whose entries are independent of $M$. For a large class of diagonal
matrices $V$, we prove that the fluctuations of linear spectral statistics of
$W$ for $C^{2}_{c}$ test function can be decomposed into that of $M$ and of
$V$, and that each of those weakly converges to a Gaussian distribution. We
also calculate the formulae for the means and variances of the limiting
distributions.
