Statistical Papers

, Volume 60, Issue 3, pp 983–1015 | Cite as

Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices

  • Ningning XiaEmail author
  • Zhidong Bai
Regular Article


In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix \(\mathbf {W}_n.\) In particular, we derive the optimal bound for the rate of convergence of the expected VESD of \(\mathbf{W}_n\) to the semicircle law, which is of order \(O(n^{-1/2})\) under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are \(O(n^{-1/4})\) and \(O(n^{-1/6}),\) 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 \(O(n^{-1/2}).\)


Wigner matrices Eigenvectors Empirical spectral distribution Semicircular law Convergence rate 



The research was supported by NSFC 11501348, Shanghai Pujiang Program 15PJ1402300, IRTSHUFE and the State Key Program in the Major Research Plan of NSFC 91546202. The research was partially supported by National Natural Science Foundation of China (Grant No. 11571067).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Statistics and Management, Shanghai Key Laboratory of Financial Information TechnologyShanghai University of Finance and EconomicsShanghaiChina
  2. 2.KLASMOE and School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina

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