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Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices

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  • Ningning Xia

    (Shanghai University of Finance and Economics)

  • Zhidong Bai

    (Northeast Normal University)

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 ) .

Suggested Citation

  • Ningning Xia & Zhidong Bai, 2019. "Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices," Statistical Papers, Springer, vol. 60(3), pages 983-1015, June.
  • Handle: RePEc:spr:stpapr:v:60:y:2019:i:3:d:10.1007_s00362-016-0860-x
    DOI: 10.1007/s00362-016-0860-x
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    References listed on IDEAS

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    1. Silverstein, Jack W., 1989. "On the eigenvectors of large dimensional sample covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 1-16, July.
    2. Weiming Li, 2014. "Local expectations of the population spectral distribution of a high-dimensional covariance matrix," Statistical Papers, Springer, vol. 55(2), pages 563-573, May.
    3. Johnstone, Iain M. & Lu, Arthur Yu, 2009. "On Consistency and Sparsity for Principal Components Analysis in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 682-693.
    4. Wenxing Guo & Xiaohui Liu & Shangli Zhang, 2016. "The principal correlation components estimator and its optimality," Statistical Papers, Springer, vol. 57(3), pages 755-779, September.
    5. Shen, Haipeng & Huang, Jianhua Z., 2008. "Sparse principal component analysis via regularized low rank matrix approximation," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1015-1034, July.
    6. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    7. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
    8. Bai, Z. D. & Miao, Baiqi & Tsay, Jhishen, 1997. "A note on the convergence rate of the spectral distributions of large random matrices," Statistics & Probability Letters, Elsevier, vol. 34(1), pages 95-101, May.
    9. Ningning Xia & Zhidong Bai, 2015. "Functional CLT of eigenvectors for large sample covariance matrices," Statistical Papers, Springer, vol. 56(1), pages 23-60, February.
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    Cited by:

    1. Li, Yuling & Zhou, Huanchao & Hu, Jiang, 2023. "The eigenvector LSD of information plus noise matrices and its application to linear regression model," Statistics & Probability Letters, Elsevier, vol. 197(C).

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