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Spectral convergence for a general class of random matrices

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  • Rubio, Francisco
  • Mestre, Xavier

Abstract

Let be an M×N complex random matrix with i.i.d. entries having mean zero and variance 1/N and consider the class of matrices of the type , where , and are Hermitian nonnegative definite matrices, such that and have bounded spectral norm with being diagonal, and is the nonnegative definite square root of . Under some assumptions on the moments of the entries of , it is proved in this paper that, for any matrix with bounded trace norm and for each complex z outside the positive real line, almost surely as M,N-->[infinity] at the same rate, where [delta]M(z) is deterministic and solely depends on and . The previous result can be particularized to the study of the limiting behavior of the Stieltjes transform as well as the eigenvectors of the random matrix model . The study is motivated by applications in the field of statistical signal processing.

Suggested Citation

  • Rubio, Francisco & Mestre, Xavier, 2011. "Spectral convergence for a general class of random matrices," Statistics & Probability Letters, Elsevier, vol. 81(5), pages 592-602, May.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:5:p:592-602
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    References listed on IDEAS

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    1. 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.
    2. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    3. Paul, Debashis & Silverstein, Jack W., 2009. "No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 37-57, January.
    4. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
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    Cited by:

    1. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2018. "Estimation of the global minimum variance portfolio in high dimensions," European Journal of Operational Research, Elsevier, vol. 266(1), pages 371-390.
    2. Taras Bodnar & Nestor Parolya & Erik Thors'en, 2022. "Two is better than one: Regularized shrinkage of large minimum variance portfolio," Papers 2202.06666, arXiv.org.
    3. Bodnar, Taras & Parolya, Nestor & Thorsén, Erik, 2023. "Is the empirical out-of-sample variance an informative risk measure for the high-dimensional portfolios?," Finance Research Letters, Elsevier, vol. 54(C).
    4. Bodnar, Taras & Dette, Holger & Parolya, Nestor, 2016. "Spectral analysis of the Moore–Penrose inverse of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 160-172.
    5. Taras Bodnar & Nestor Parolya & Erik Thorsen, 2021. "Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio," Papers 2106.02131, arXiv.org, revised Nov 2021.
    6. Taras Bodnar & Nikolaus Hautsch & Yarema Okhrin & Nestor Parolya, 2024. "Consistent Estimation of the High-Dimensional Efficient Frontier," Papers 2409.15103, arXiv.org.
    7. Taras Bodnar & Yarema Okhrin & Nestor Parolya, 2022. "Optimal Shrinkage-Based Portfolio Selection in High Dimensions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(1), pages 140-156, December.
    8. Taras Bodnar & Arjun K. Gupta & Nestor Parolya, 2013. "Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix," Papers 1308.0931, arXiv.org, revised Mar 2014.
    9. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2016. "Direct shrinkage estimation of large dimensional precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 223-236.
    10. Francisco Rubio & Xavier Mestre & Daniel P. Palomar, 2011. "Performance analysis and optimal selection of large mean-variance portfolios under estimation risk," Papers 1110.3460, arXiv.org.
    11. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2014. "On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 215-228.
    12. Bodnar, Olha & Bodnar, Taras & Parolya, Nestor, 2022. "Recent advances in shrinkage-based high-dimensional inference," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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