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Spectral Norm of Circulant-Type Matrices

Author

Listed:
  • Arup Bose

    (Indian Statistical Institute)

  • Rajat Subhra Hazra

    (Indian Statistical Institute)

  • Koushik Saha

    (Indian Statistical Institute)

Abstract

We first discuss the convergence in probability and in distribution of the spectral norm of scaled Toeplitz, circulant, reverse circulant, symmetric circulant, and a class of k-circulant matrices when the input sequence is independent and identically distributed with finite moments of suitable order and the dimension of the matrix tends to ∞. When the input sequence is a stationary two-sided moving average process of infinite order, it is difficult to derive the limiting distribution of the spectral norm, but if the eigenvalues are scaled by the spectral density, then the limits of the maximum of modulus of these scaled eigenvalues can be derived in most of the cases.

Suggested Citation

  • Arup Bose & Rajat Subhra Hazra & Koushik Saha, 2011. "Spectral Norm of Circulant-Type Matrices," Journal of Theoretical Probability, Springer, vol. 24(2), pages 479-516, June.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:2:d:10.1007_s10959-009-0257-z
    DOI: 10.1007/s10959-009-0257-z
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    References listed on IDEAS

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    1. Christopher Hammond & Steven J. Miller, 2005. "Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices," Journal of Theoretical Probability, Springer, vol. 18(3), pages 537-566, July.
    2. Einmahl, Uwe, 1989. "Extensions of results of Komlós, Major, and Tusnády to the multivariate case," Journal of Multivariate Analysis, Elsevier, vol. 28(1), pages 20-68, January.
    3. Bose, Arup & Mitra, Joydip, 2002. "Limiting spectral distribution of a special circulant," Statistics & Probability Letters, Elsevier, vol. 60(1), pages 111-120, November.
    4. Ming Dai & Arunava Mukherjea, 2001. "Identification of the Parameters of a Multivariate Normal Vector by the Distribution of the Maximum," Journal of Theoretical Probability, Springer, vol. 14(1), pages 267-298, January.
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