Fluctuations for linear eigenvalue statistics of sample covariance matrices

2018/06/22 by Cipolloni, Giorgio, Erdős, László
#15B52 #60B20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1806.08751

Abstract

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix \widetildeW and its minor W. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of \widetildeW and W. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices the fluctuation may entirely vanish.

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