Necessary and Sufficient Conditions for Convergence to the Semicircle Distribution

TL;DR

This paper provides necessary and sufficient conditions for the convergence of spectral distributions of Hermitian matrices to the semicircle law, extending classical results to matrices with infinite variance entries.

Contribution

It characterizes convergence to the semicircle law based on entry variances and extends the theory to matrices with infinite second moments.

Findings

Convergence to the semicircle law is characterized by variance conditions.

The results include matrices with infinite second moments.

Convergence in distribution of row sums to normal is equivalent to spectral convergence.

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.