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A Note on the Convergence of the Extreme Eigenvalues of a Large-Dimensional Sample Covariance Matrix

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  • Shizhe Hong

    (Shanghai University of Finance and Economics)

  • Haiyan Song

    (Northeast Normal University)

  • Jiang Hu

    (Northeast Normal University)

  • Zhidong Bai

    (Northeast Normal University
    Xi’an Jiaotong University)

Abstract

In this study, we explore both weak and strong convergence properties of extreme eigenvalues in a large-dimensional sample covariance matrix, specifically in cases where the data matrix comprises independent, though not identically distributed, elements. Our findings reveal that, provided there exists a uniform boundedness condition on the $$(2+\delta )$$ ( 2 + δ ) -th moment for some $$\delta >0$$ δ > 0 and the proper Lindeberg condition is satisfied, the established convergence results in Yin, Bai and Krishnaiah, (Probab. Theory Relat. Fields 78:509–521, 1988) and Bai, and Yin (Ann. Probab. 21:1275–1294, 1993) remain applicable.

Suggested Citation

  • Shizhe Hong & Haiyan Song & Jiang Hu & Zhidong Bai, 2025. "A Note on the Convergence of the Extreme Eigenvalues of a Large-Dimensional Sample Covariance Matrix," Journal of Theoretical Probability, Springer, vol. 38(2), pages 1-16, June.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:2:d:10.1007_s10959-025-01415-y
    DOI: 10.1007/s10959-025-01415-y
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    References listed on IDEAS

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    1. Bai, Z. D. & Silverstein, Jack W. & Yin, Y. Q., 1988. "A note on the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 166-168, August.
    2. Forzani, Liliana & Gieco, Antonella & Tolmasky, Carlos, 2017. "Likelihood ratio test for partial sphericity in high and ultra-high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 18-38.
    3. 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.
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