Some estimates for generalized Wigner matrix linear spectral statistics

TL;DR

This paper derives precise expansions for the characteristic function of linear spectral statistics of generalized Wigner matrices, revealing Gaussian behavior and non-Gaussian corrections with improved error estimates over previous results.

Contribution

It extends existing spectral statistic expansions to generalized Wigner matrices with sharper error bounds and identifies non-Gaussian corrections.

Findings

Expansion of characteristic function with error $ ext{O}(N^{-1})$

Identification of non-Gaussian corrections of size $ ext{O}(N^{-1/2})$

Applications demonstrating the theoretical results

Abstract

We consider the characteristic function of linear spectral statistics of generalized Wigner matrices. We provide an expansion of the characteristic function with error $\mathcal{O} ( N^{-1})$ around its limiting Gaussian form, and identify sub-leading non-Gaussian corrections of size $\mathcal{O} (N^{-1/2})$. Prior expansions with this error rate held only for Wigner matrices; only a weaker error rate was available for more general matrix ensembles. We provide some applications.