Abstract
We consider the eigenvalues of sample covariance matrices of the form . The sample X is an rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. Assuming that the limiting spectral density of Σ exhibits convex decay at the right edge of the spectrum, in the limit with , we find a certain threshold such that for the limiting spectral distribution of also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of are determined by the order statistics of the eigenvalues of Σ, and in particular, the limiting distribution of the largest eigenvalue of is given by a Weibull distribution. In case , we also prove that the limiting distribution of the largest eigenvalue of is Gaussian if the entries of Σ are i.i.d. random variables. While Σ is considered to be random mostly, the results also hold for deterministic Σ with some additional assumptions.
Acknowledgements
We thank Paul Jung for helpful discussions. The work of J. Kwak was partially supported by National Research Foundation of Korea under grant number NRF-2017R1A2B2001952. The work of J. O. Lee and J. Park was partially supported by National Research Foundation of Korea under grant number NRF-2019R1A5A1028324.
Citation
Jinwoong Kwak. Ji Oon Lee. Jaewhi Park. "Extremal eigenvalues of sample covariance matrices with general population." Bernoulli 27 (4) 2740 - 2765, November 2021. https://doi.org/10.3150/21-BEJ1329
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