Necessary and Sufficient Conditions for Convergence to the Semicircle Distribution

Abstract: 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 the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain matrices with entries having infinite second moments. As a corollary, another characterization of semicircle convergence is given in terms of convergence in distribution of the row sums to the standard normal distribution.