Transferring supremum-norm rates and weak convergence of covariance kernel estimators to functional principal components

We show that $L_2$-perturbation theory can be used to transfer rates of convergence in the supremum norm as well as weak convergence in the space of continuous functions from covariance kernel estimators to the associated functional principle components (FPCs). As an application we obtain optimal rates of convergence in sup-norm, including minimax-lower bounds, as well as asymptotic normality for estimating the FPCs in a discrete observational model with errors under fixed, synchronous design. The sparse to dense transition which has previously been observed for mean function and covariance kernel estimators also applies to the FPCs. Surprisingly, eigenvalue estimation exhibits a discretization-dominated regime under sparse designs, too. Our results further apply to estimators of cross-covariance and long-run covariance kernels, as well as to covariance kernels of derivative processes. We also present results of numerical experiments in which we use the Nystr\"om method to compute FPCs and eigenvalues, and give an empirical illustration to series of daily temperature curves.