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On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix

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  • Silverstein, Jack W.

Abstract

Let {wij}, i, J = 1, 2, ..., be i.i.d. random variables and for each n let Mn = (1/n) WnWnT, where Wn = (wij), i = 1, 2, ..., p; j = 1, 2, ..., n; p = p(n), and p/n --> y > 0 as n --> [infinity]. The weak behavior of the largest eigenvalue of Mn is studied. The primary aim of the paper is to show that the largest eigenvalue converges in probability to a nonrandom quantity if and only if E(w11) = 0 and n4P([omega]11 >= n) = o(1), the limit being (1 + [radical sign]y)2 E(w112).

Suggested Citation

  • Silverstein, Jack W., 1989. "On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 30(2), pages 307-311, August.
    • Handle: RePEc:eee:jmvana:v:30:y:1989:i:2:p:307-311
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    Citations

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    Cited by:

    1. Hugo Freeman & Martin Weidner, 2021. "Low-rank approximations of nonseparable panel models," The Econometrics Journal, Royal Economic Society, vol. 24(2), pages 40-77.
    2. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    3. Li, Yuling & Zhou, Huanchao & Hu, Jiang, 2023. "The eigenvector LSD of information plus noise matrices and its application to linear regression model," Statistics & Probability Letters, Elsevier, vol. 197(C).
    4. Elizaveta Rebrova, 2020. "Constructive Regularization of the Random Matrix Norm," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1768-1790, September.
    5. Bai, Zhidong & Silverstein, Jack W., 2022. "A tribute to P.R. Krishnaiah," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    6. Hyungsik Roger Moon & Martin Weidner, 2015. "Linear Regression for Panel With Unknown Number of Factors as Interactive Fixed Effects," Econometrica, Econometric Society, vol. 83(4), pages 1543-1579, July.
    7. Xu, Yangchang & Xia, Ningning, 2023. "On the eigenvectors of large-dimensional sample spatial sign covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    8. Hyungsik Roger Roger Moon & Martin Weidner, 2013. "Linear regression for panel with unknown number of factors as interactive fixed effects," CeMMAP working papers 49/13, Institute for Fiscal Studies.
    9. Hyungsik Roger Roger Moon & Martin Weidner, 2014. "Dynamic linear panel regression models with interactive fixed effects," CeMMAP working papers 47/14, Institute for Fiscal Studies.
    10. Hyungsik Roger Roger Moon & Martin Weidner, 2014. "Linear regression for panel with unknown number of factors as interactive fixed effects," CeMMAP working papers 35/14, Institute for Fiscal Studies.
    11. Moon, Hyungsik Roger & Weidner, Martin, 2017. "Dynamic Linear Panel Regression Models With Interactive Fixed Effects," Econometric Theory, Cambridge University Press, vol. 33(1), pages 158-195, February.
    12. Hyungsik Roger Roger Moon & Martin Weidner, 2013. "Dynamic linear panel regression models with interactive fixed effects," CeMMAP working papers 63/13, Institute for Fiscal Studies.
    13. Iv'an Fern'andez-Val & Hugo Freeman & Martin Weidner, 2020. "Low-Rank Approximations of Nonseparable Panel Models," Papers 2010.12439, arXiv.org, revised Mar 2021.
    14. Ningning Xia & Zhidong Bai, 2015. "Functional CLT of eigenvectors for large sample covariance matrices," Statistical Papers, Springer, vol. 56(1), pages 23-60, February.

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