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

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 $\epsilon >0$.

arXiv commentThis paper is an addendum to arXiv:1405.7820 , relaxing moment conditions in Theorem 1.3 of that paper to order $4+\epsilon$. For a discussion of the relevant literature we refer to the previous paper arXiv:1405.7820 (to appear Probab. Theory Relat. Fields (2015+))