On Cramér–von Mises statistic for the spectral distribution of random matrices

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Let FN and F be the empirical and limiting spectral distributions of an N×N Wigner matrix. The Cramér–von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between FN and F in L2-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.