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Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

Author

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  • Yi-Ting Li

    (Peking University)

  • Dang-Zheng Liu

    (Peking University)

  • Zheng-Dong Wang

    (Peking University)

Abstract

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulas of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit $\gamma_{T}^{(m)}$ $(\gamma_{H}^{(m)})$ of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where m is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that $\gamma_{T}^{(m)}$ $(\gamma_{H}^{(m)})$ converges weakly to the semicircle law as m→∞.

Suggested Citation

  • Yi-Ting Li & Dang-Zheng Liu & Zheng-Dong Wang, 2011. "Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices," Journal of Theoretical Probability, Springer, vol. 24(4), pages 1063-1086, December.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-010-0326-3
    DOI: 10.1007/s10959-010-0326-3
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    Cited by:

    1. Philippe Loubaton, 2016. "On the Almost Sure Location of the Singular Values of Certain Gaussian Block-Hankel Large Random Matrices," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1339-1443, December.
    2. Debapratim Banerjee & Arup Bose, 2016. "Bulk behaviour of some patterned block matrices," Indian Journal of Pure and Applied Mathematics, Springer, vol. 47(2), pages 273-289, June.

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