Abstract
Let and F be the empirical and limiting spectral distributions of an Wigner matrix. The Cramér–von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between and F in -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.
Funding Statement
Z. G. Bao was partially supported by Hong Kong RGC Grant GRF 16300618, GRF 16301520 and GRF 16305421. Y. K. He was partially supported by NCCR Swissmap, SNF Grant No. 20020_1726, and CityU Start-up Grant No. 7200727.
Acknowledgments
We would like to thank Jiang Hu for discussion and simulation. We would also like to thank Yan Fyodorov, Gaultier Lambert, Dong Wang and Lun Zhang for reference and helpful comments. Finally, we are grateful to the anonymous referee for useful remarks and comments.
Citation
Zhigang Bao. Yukun He. "On Cramér–von Mises statistic for the spectral distribution of random matrices." Ann. Appl. Probab. 32 (6) 4315 - 4355, December 2022. https://doi.org/10.1214/22-AAP1788
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