Necessary and sufficient conditions for convergence to the semicircle distribution

We consider random Hermitian matrices with independent upper triangular entries. Wigner’s semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to distribution. characterize convergence in terms of variances entries, natural assumptions such as Lindeberg condition. The result extends entries having infinite second moments. As a corollary, another characterization is given row sums standard normal