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Limiting spectral distribution of block matrices with Toeplitz block structure

Author

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  • Basu, Riddhipratim
  • Bose, Arup
  • Ganguly, Shirshendu
  • Hazra, Rajat Subhra

Abstract

We study two specific symmetric random block Toeplitz (of dimension k×k) matrices, where the blocks (of size n×n) are (i) matrices with i.i.d. entries and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by nk) when (a) k is fixed and n→∞ (b) n is fixed and k→∞ (c) n and k go to ∞ simultaneously. Further, the LSDs obtained in (a) and (b) coincide with those in (c) when n or respectively k tends to infinity. This limit in (c) is the semicircle law in Case (i). In Case (ii), the limit is related to the limit of the random symmetric Toeplitz matrix as obtained by Bryc et al. (2006) and Hammond and Miller (2005).

Suggested Citation

  • Basu, Riddhipratim & Bose, Arup & Ganguly, Shirshendu & Hazra, Rajat Subhra, 2012. "Limiting spectral distribution of block matrices with Toeplitz block structure," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1430-1438.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:7:p:1430-1438
    DOI: 10.1016/j.spl.2012.04.004
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    Cited by:

    1. Ding, Xue, 2015. "On some spectral properties of large block Laplacian random matrices," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 61-69.
    2. 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.
    3. 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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