New estimators of spectral distributions of Wigner matrices

TL;DR

This paper introduces kernel estimators for the spectral distribution of Wigner matrices, demonstrating their consistency and effectiveness, even when matrix elements lack finite variance but are in the domain of attraction of the normal law.

Contribution

It presents novel kernel estimators for Wigner matrices' spectral distributions and proves their consistency, extending applicability beyond finite variance cases.

Findings

Kernel estimators are consistent for Wigner spectral distributions.

Performance of estimators confirmed through simulation studies.

Semicircle law holds for new estimators even with infinite variance elements.

Abstract

We introduce kernel estimators for the semicircle law. In this first part of our general theory on the estimators, we prove the consistency and conduct simulation study to show the performance of the estimators. We also point out that Wigner's semicircle law for our new estimators and the classical empirical spectral distributions is still true when the elements of Wigner matrices don't have finite variances but are in the domain of attraction of normal law.