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Bulk behaviour of some patterned block matrices

Author

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  • Debapratim Banerjee

    (Indian Statistical Institute)

  • Arup Bose

    (Indian Statistical Institute)

Abstract

We investigate the bulk behaviour of singular values and/or eigenvalues of two types of block random matrices. In the first one, we allow unrestricted structure of order m × p with n × n blocks and in the second one we allow m × m Wigner structure with symmetric n × n blocks. Different rows of blocks are assumed to be independent while the blocks within any row satisfy a weak dependence assumption that allows for some repetition of random variables among nearby blocks. In general, n can be finite or can grow to infinity. Suppose the input random variables are i.i.d. with mean 0 and variance 1 with finite moments of all orders. We prove that under certain conditions, the Marčenko-Pastur result holds in the first model when m → ∞ and $$\tfrac{m} {p} \to c \in (0,\infty )$$ , and the semicircular result holds in the second model when m → ∞. These in particular generalize the bulk behaviour results of Loubaton [10].

Suggested Citation

  • 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.
    • Handle: RePEc:spr:indpam:v:47:y:2016:i:2:d:10.1007_s13226-016-0187-2
    DOI: 10.1007/s13226-016-0187-2
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    References listed on IDEAS

    as
    1. 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.
    2. Sayan Banerjee & Arup Bose, 2013. "Noncrossing Partitions, Catalan Words, and the Semicircle Law," Journal of Theoretical Probability, Springer, vol. 26(2), pages 386-409, June.
    3. 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.
    Full references (including those not matched with items on IDEAS)

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

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