On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications
Abstract
We study the eigenvector-weighted spectral statistics Tn(φ,g)=p−1tr{φ(Sn)g(Σ)} of large-dimensional sample covariance matrices Sn with population covariance Σ. These statistics combine functions of sample eigenvalues with sample--population eigenvector overlaps and arise in statistical losses that depend on both. As the dimension and sample size grow proportionally, we establish a joint central limit theorem for {Tn(φa,ga)}a=1r with multiple analytic test functions and population weights, centered by finite-dimensional deterministic equivalents. A key intermediate step is a joint Gaussian process limit for the weighted resolvent traces mng(z)=p−1tr{(Sn−zIp)−1g(Σ)}. The limiting means and cross-covariances are expressed in terms of the population spectral distribution. We apply these results in two settings: one concerns the variance component of minimum-norm prediction risk, and the other concerns the Frobenius loss of fixed rotationally invariant covariance estimators. For fixed affine shrinkage rules, the loss means, variances, and cross-covariances have explicit formulas in terms of population spectral moments.
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BibTeX
@misc{li2026eigenvector,
title = {{On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications}},
author = {Weiming Li and Zeng Li and Qinwen Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12928},
eprint = {2610.12928},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12928},
}
RIS
TY - UNPB
TI - On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications
AU - Weiming Li
AU - Zeng Li
AU - Qinwen Wang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12928
UR - https://arxiv.org/abs/2610.12928
AB - We study the eigenvector-weighted spectral statistics T_n(φ,g)=p^−1tr{φ(S_n)g(Σ)} of large-dimensional sample covariance matrices S_n with population covariance Σ. These statistics combine functions of sample eigenvalues with sample--population eigenvector overlaps and arise in statistical losses that depend on both. As the dimension and sample size grow proportionally, we establish a joint central limit theorem for {T_n(φ_a,g_a)}_a=1^r with multiple analytic test functions and population weights, centered by finite-dimensional deterministic equivalents. A key intermediate step is a joint Gaussian process limit for the weighted resolvent traces m_n^g(z)=p^−1tr{(S_n−zI_p)^−1g(Σ)}. The limiting means and cross-covariances are expressed in terms of the population spectral distribution. We apply these results in two settings: one concerns the variance component of minimum-norm prediction risk, and the other concerns the Frobenius loss of fixed rotationally invariant covariance estimators. For fixed affine shrinkage rules, the loss means, variances, and cross-covariances have explicit formulas in terms of population spectral moments.
ER -