Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices

Ningning Xia () and Zhidong Bai ()
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Ningning Xia: Shanghai University of Finance and Economics
Zhidong Bai: Northeast Normal University

Statistical Papers, 2019, vol. 60, issue 3, No 17, 983-1015

Abstract: Abstract In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix $$\mathbf {W}_n.$$ W n . In particular, we derive the optimal bound for the rate of convergence of the expected VESD of $$\mathbf{W}_n$$ W n to the semicircle law, which is of order $$O(n^{-1/2})$$ O ( n - 1 / 2 ) under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are $$O(n^{-1/4})$$ O ( n - 1 / 4 ) and $$O(n^{-1/6}),$$ O ( n - 1 / 6 ) , respectively, under finite eighth moment condition. Numerical studies demonstrate that the convergence rate does not depend on the choice of unit vector involved in the VESD function, and the best possible bound for the rate of convergence of the VESD is of order $$O(n^{-1/2}).$$ O ( n - 1 / 2 ) .

Keywords: Wigner matrices; Eigenvectors; Empirical spectral distribution; Semicircular law; Convergence rate (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s00362-016-0860-x

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