  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.

Probability