On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications

We study the eigenvector-weighted spectral statistics $$T_n(\phi ,g)=p^{-1}tr\{\phi ({\mathit{S}}_n)g(\mathbf{\Sigma })\}$$ of large-dimensional sample covariance matrices ${\mathit{S}}_n$ with population covariance $\mathbf{\Sigma }$. 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 $\{T_n({\phi }_a,g_a){\}}_{a=1}^r$ 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 $$m_n^g(z)=p^{-1}tr\{({\mathit{S}}_n-z{\mathit{I}}_p{)}^{-1}g(\mathbf{\Sigma })\}\mathrm{.}$$ 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.