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CLT for spiked eigenvalues of a sample covariance matrix from high-dimensional Gaussian mean mixtures

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  • Li, Weiming
  • Zhu, Junpeng

Abstract

Sample covariance matrices from a finite mean mixture model naturally carry certain spiked eigenvalues, which are generated by the differences among the mean vectors. However, their asymptotic behaviors remain largely unknown when the population dimension p grows proportionally to the sample size n. In this paper, a new CLT is established for the spiked eigenvalues by considering a Gaussian mean mixture in such high-dimensional asymptotic frameworks. It shows that the convergence rate of these eigenvalues is O(1/n) and their fluctuations can be characterized by the mixing proportions, the eigenvalues of the common covariance matrix, and the inner products between the mean vectors and the eigenvectors of the covariance matrix.

Suggested Citation

  • Li, Weiming & Zhu, Junpeng, 2023. "CLT for spiked eigenvalues of a sample covariance matrix from high-dimensional Gaussian mean mixtures," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:jmvana:v:193:y:2023:i:c:s0047259x2200118x
    DOI: 10.1016/j.jmva.2022.105127
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    References listed on IDEAS

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    1. 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.
    2. Bai, Zhidong & Yao, Jianfeng, 2012. "On sample eigenvalues in a generalized spiked population model," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 167-177.
    3. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
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