Limit theorems for two classes of random matrices with Gaussian elements

In this note, we consider ensembles of random symmetric matrices with Gaussian elements. Suppose that $\mathbb EX_{ij}=0$ and $\mathbb EX_{ij}^2=\sigma_{ij}^2$. We do not assume that all $\sigma_{ij}$ are equal. Assuming that the average of the normalized sums of variances in each row converges to one and Lindeberg condition holds true we prove that the empirical spectral distribution of eigenvalues converges to Wigner's semicircle law. We also provide analogue of this result for sample covariance matrices and prove convergence to the Marchenko–Pastur law.