# An Exposition on Wigner's Semicircular Law

## TL;DR

This paper refines the semicircular law for Hermitian random matrices using the moment method, accommodating broader conditions including infinite variance cases, and provides detailed expositions suitable for newcomers.

## Contribution

It introduces a strengthened semicircular law under weaker assumptions and extends results to cases with row sums converging to normal distribution, including infinite variance scenarios.

## Key findings

- Strengthened semicircular law under minimal assumptions
- Extension to matrices with infinite variance entries
- Detailed exposition suitable for beginners

## Abstract

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.