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A New Method for Bounding Rates of Convergence of Empirical Spectral Distributions

Author

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  • S. Chatterjee

    (Stanford University)

  • A. Bose

    (Theoretical Statistics and Mathematics Unit)

Abstract

The probabilistic properties of eigenvalues of random matrices whose dimension increases indefinitely has received considerable attention. One important aspect is the existence and identification of the limiting spectral distribution (LSD) of the empirical distribution of the eigenvalues. When the LSD exists, it is useful to know the rate at which the convergence holds. The main method to establish such rates is the use of Stieltjes transform. In this article we introduce a new technique of bounding the rates of convergence to the LSD. We show how our results apply to specific cases such as the Wigner matrix and the Sample Covariance matrix.

Suggested Citation

  • S. Chatterjee & A. Bose, 2004. "A New Method for Bounding Rates of Convergence of Empirical Spectral Distributions," Journal of Theoretical Probability, Springer, vol. 17(4), pages 1003-1019, October.
    • Handle: RePEc:spr:jotpro:v:17:y:2004:i:4:d:10.1007_s10959-004-0587-9
    DOI: 10.1007/s10959-004-0587-9
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    References listed on IDEAS

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    1. Bose, Arup & Mitra, Joydip, 2002. "Limiting spectral distribution of a special circulant," Statistics & Probability Letters, Elsevier, vol. 60(1), pages 111-120, November.
    2. Yin, Y. Q., 1986. "Limiting spectral distribution for a class of random matrices," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 50-68, October.
    3. Jonsson, Dag, 1982. "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 1-38, March.
    4. 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.
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    Cited by:

    1. S. G. Bobkov & F. Götze & A. N. Tikhomirov, 2010. "On Concentration of Empirical Measures and Convergence to the Semi-circle Law," Journal of Theoretical Probability, Springer, vol. 23(3), pages 792-823, September.

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