Title
The Wilcoxon--Wigner Random Matrix of Normalized Rank Statistics.
Abstract
This paper studies large symmetric matrices, called Wilcoxon--Wigner random matrices, whose entries are normalized rank statistics derived from an underlying i.i.d. sample of continuous random variables. These matrices naturally arise as the matricization of one-sample problems and conceptually lie at the intersection of nonparametrics, multivariate analysis, and dimension reduction. Despite having discrete, dependent entries, we show that their spectra exhibit the classical Wigner semicircle law and Bai--Yin-type operator norm convergence after suitable centering and scaling, thereby matching the behavior of classical ensembles of symmetric random matrices with independent entries. We further show that the outlier eigenvalue and its eigenvector admit asymptotically Gaussian fluctuations, as with independent entry models, but with a different scaling and asymptotic variance that together highlight the dependent, constrained nature of the normalized rank statistics.