Optimal bounds for convergence of expected spectral distributions to the semi-circular law for the $4+\epsilon$ moment ensemble

TL;DR

This paper improves the understanding of how quickly the expected spectral distribution of Wigner matrices converges to the semicircular law by relaxing moment conditions from order 8 to just above 4.

Contribution

It extends previous bounds on convergence rates to Wigner matrices by lowering the required moment condition from 8 to 4+epsilon, broadening applicability.

Findings

Established convergence bounds under weaker moment conditions

Extended previous $O(n^{-1})$ convergence rate results

Applicable to a wider class of Wigner matrices

Abstract

This paper extends a previous bound of order $O(n^{-1})$ of the authors (arXiv:1405.7820[math.PR]), for the rate of convergence in Kolmogorov distance of the expected spectral distribution of a Wigner random matrix ensemble to the semicircular law. Here we relax the moment conditions for entries of the Wigner matrices from order $8$ to order $4+ \epsilon$ for an arbitrary small $\epsilon>0$.