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

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 ${\mathrm{\ell }}^2$-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.