On Cram\'{e}r-von Mises statistic for the spectral distribution of random matrices

TL;DR

This paper investigates the mesoscopic approximation of the Cramér-von Mises statistic for spectral distributions of Wigner matrices, deriving its limiting distribution and comparing it with the classical case for CUE.

Contribution

It introduces a mesoscopic approximation of the CvM statistic for Wigner matrices and derives its limiting distribution, extending classical goodness-of-fit analysis to random matrix spectra.

Findings

Derived the limiting distribution of the mesoscopic CvM statistic for Wigner matrices.

Provided the limiting distribution of the classical CvM statistic for the CUE toy model.

Extended goodness-of-fit testing methods to spectral distributions of random matrices.

Abstract

Let $F_N$ and $F$ be the empirical and limiting spectral distributions of an $N\times N$ Wigner matrix. The Cram\'{e}r-von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between $F_N$ and $F$ in $\ell^2$-norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE.