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

Shizhe Hong, Haiyan Song (), Jiang Hu and Zhidong Bai
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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

Journal of Theoretical Probability, 2025, vol. 38, issue 2, 1-16

Abstract: 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.

Keywords: Extreme eigenvalues; Large dimension; Sample covariance matrix; 60B20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01415-y

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