  We revisit the moment method to obtain a slightly strengthened version of the
usual semicircular law. Our version assumes only that the upper triangular
entries of Hermitian random matrices are independent, have mean zero and
variances close to $1/n$ in a certain sense, and satisfy a Lindeberg-type
condition. As an application, we derive another semicircular law for the case
when the sum of a row converges in distribution to the standard normal
distribution, including the case where all matrix entries may have infinite
variance. The appendix, making up the majority of the paper, provides for those
new to the subject, a rigorous exposition of most details involved, including
also a proof of a semicircular law that uses the Stieltjes transform method.
