Theory in functional principal components of discretely observed data

Convergence of eigenfunctions with diverging index is essential in nearly all methods based on functional principal components analysis. The main goal of this work is to establish the unified theory for such eigencomponents in different types of convergence based on discretely observed functional data. We obtain the moment bounds for eigenfunctions and eigenvalues for a wide range of the sampling rate and show that under some mild assumptions, the $\mathcal{L}^{2}$ bound of eigenfunctions estimator with diverging indices is optimal in the minimax sense as if the curves are fully observed. This is the first attempt at obtaining an optimal rate for eigenfunctions with diverging index for discretely observed functional data. We propose a double truncation technique in handling the uniform convergence of function data and establish the uniform convergence of covariance function as well as the eigenfunctions for all sampling scheme under mild assumptions. The technique route proposed in this work provides a new tool in handling the perturbation series with discretely observed functional data and can be applied in most problems based on functional principal components analysis and models involving inverse issue.