Concentration and consistency of sample covariance matrix functionals in sub-Gaussian models

We establish nonasymptotic concentration results for a broad set of functionals of sample covariance matrices in sub-Gaussian models that are given in terms of the effective rank r(Σ) of the population covariance matrix Σ and lay the groundwork for statistical analysis in a dimension-free framework without structural or sparsity assumptions on Σ. The obtained rates of convergence and error bounds for the functionals incorporate the dependence on r(Σ) known for the sample covariance matrix itself and ensure consistency of the plug-in estimators of covariance matrix functionals under the same asymptotic behavior of r(Σ) as in the case of plug-in estimators of Σ. The admissible types of functionals include the linear spectral statistic arising from functions with bounded Lipschitz derivatives as well as the bilinear form arising from Lipschitz functions and, when Σ is invertible, from certain non-Lipschitz functions as well. Concentration and consistency results in terms of r(Σ) are also established for spectral measures of sample covariance matrices.