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

Abstract

We study the eigenvector-weighted spectral statistics \[ T_n(φ,g)=p^{-1}tr\{φ(\boldsymbol{S}_n)g(\boldsymbolΣ)\} \] of large-dimensional sample covariance matrices \(\boldsymbol{S}_n\) with population covariance \(\boldsymbolΣ\). 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 \[ m_n^g(z)=p^{-1}tr\{(\boldsymbol{S}_n-z\boldsymbol{I}_p)^{-1}g(\boldsymbolΣ)\}. \] 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.