Law of large numbers for discretely observed random functions

A strong law of large numbers for continuous random functions, and associated tensor product surfaces is established in the setup of discretely observed functional data. The result is shown in the framework of uniform convergence of functions, and stated without imposing any distributional assumptions. It is demonstrated that, under mild conditions, laws of large numbers for continuously observed functional data imply the corresponding laws under the discrete observational design of functions. Applications to the problem of estimation of expectation functions and covariance surfaces for discretely observed functional data are discussed.