Consider the sample covariance matrix (\Sigma^{1/2})XX^*(\Sigma^{1/2}) where X is an M by N random matrix with independent entries and \Sigma is an M by M positive definite diagonal matrix. Use L(f) to denote the linear spectral statistics of the sample covariance matrix with test function f. It is known that if \Sigma is deterministic, then the fluctuation of L(f) converges in distribution to a Gaussian distribution. We prove that if \Sigma is random and is independent of X, then L(f) multiplied by N^{-1/2} converges in distribution to a Gaussian distribution. This phenomenon implies that the randomness of \Sigma weakens the correlation among the eigenvalues of the sample covariance matrix. This is a joint work with Ji Oon Lee.