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Some limit theorems for the eigenvalues of a sample covariance matrix

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  • Jonsson, Dag

Abstract

Limit theorems are given for the eigenvalues of a sample covariance matrix when the dimension of the matrix as well as the sample size tend to infinity. The limit of the cumulative distribution function of the eigenvalues is determined by use of a method of moments. The proof is mainly combinatorial. By a variant of the method of moments it is shown that the sum of the eigenvalues, raised to k-th power, k = 1, 2,..., m is asymptotically normal. A limit theorem for the log sum of the eigenvalues is completed with estimates of expected value and variance and with bounds of Berry-Esseen type.

Suggested Citation

  • Jonsson, Dag, 1982. "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 1-38, March.
    • Handle: RePEc:eee:jmvana:v:12:y:1982:i:1:p:1-38
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    Cited by:

    1. Paolo Bartesaghi & Fernando Diaz-Diaz & Rosanna Grassi & Pierpaolo Uberti, 2024. "Global Balance and Systemic Risk in Financial Correlation Networks," Papers 2407.14272, arXiv.org.
    2. Srivastava, Muni S. & Kollo, Tõnu & von Rosen, Dietrich, 2011. "Some tests for the covariance matrix with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1090-1103, July.
    3. S. Chatterjee & A. Bose, 2004. "A New Method for Bounding Rates of Convergence of Empirical Spectral Distributions," Journal of Theoretical Probability, Springer, vol. 17(4), pages 1003-1019, October.
    4. Banna, Marwa & Najim, Jamal & Yao, Jianfeng, 2020. "A CLT for linear spectral statistics of large random information-plus-noise matrices," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2250-2281.
    5. Chen, Jiaqi & Zhang, Yangchun & Li, Weiming & Tian, Boping, 2018. "A supplement on CLT for LSS under a large dimensional generalized spiked covariance model," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 57-65.
    6. Dörnemann, Nina & Dette, Holger, 2023. "Fluctuations of the diagonal entries of a large sample precision matrix," Statistics & Probability Letters, Elsevier, vol. 198(C).
    7. Bai, Zhidong & Silverstein, Jack W., 2022. "A tribute to P.R. Krishnaiah," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    8. Jan Nagel, 2021. "A Functional CLT for Partial Traces of Random Matrices," Journal of Theoretical Probability, Springer, vol. 34(2), pages 953-974, June.
    9. Yao, Jianfeng, 2012. "A note on a Marčenko–Pastur type theorem for time series," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 22-28.
    10. Mansoor Sheikh & A. C. C. Coolen, 2020. "Accurate Bayesian Data Classification Without Hyperparameter Cross-Validation," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 277-297, July.
    11. Ningning Xia & Zhidong Bai, 2015. "Functional CLT of eigenvectors for large sample covariance matrices," Statistical Papers, Springer, vol. 56(1), pages 23-60, February.
    12. Pan, Guangming & Miao, Baiqi & Jin, Baisuo, 2008. "Central limit theorem of random quadratics forms involving random matrices," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 804-809, April.
    13. Birke, Melanie & Dette, Holger, 2003. "A note on testing the covariance matrix for large dimension," Technical Reports 2004,02, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    14. Bender, Martin, 2008. "Global fluctuations in general [beta] Dyson's Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1022-1042, June.
    15. Adhikari, Kartick & Saha, Koushik, 2018. "Universality in the fluctuation of eigenvalues of random circulant matrices," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 1-8.
    16. 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.
    17. Cai, T. Tony & Liang, Tengyuan & Zhou, Harrison H., 2015. "Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 161-172.
    18. Birke, Melanie & Dette, Holger, 2005. "A note on testing the covariance matrix for large dimension," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 281-289, October.
    19. Yu, Philip L.H. & Wang, Xiaohang & Zhu, Yuanyuan, 2017. "High dimensional covariance matrix estimation by penalizing the matrix-logarithm transformed likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 12-25.
    20. Bai, Zhidong & Liu, Huixia & Wong, Wing-Keung, 2016. "Making Markowitz's Portfolio Optimization Theory Practically Useful," MPRA Paper 74360, University Library of Munich, Germany.
    21. Tingting Zou & Shurong Zheng & Zhidong Bai & Jianfeng Yao & Hongtu Zhu, 2022. "CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data," Statistical Papers, Springer, vol. 63(2), pages 605-664, April.
    22. Friesen, Olga & Löwe, Matthias & Stolz, Michael, 2013. "Gaussian fluctuations for sample covariance matrices with dependent data," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 270-287.
    23. Klein, Daniel & Pielaszkiewicz, Jolanta & Filipiak, Katarzyna, 2022. "Approximate normality in testing an exchangeable covariance structure under large- and high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 192(C).

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