Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties

We establish a Karhunen-Loève expansion for generic centered, second order stochastic processes, which does not rely on topological assumptions. We further investigate in which norms the expansion converges and derive exact average rates of convergence for these norms. For Gaussian processes as well as for some other processes we additionally prove certain sharpness results in terms of the norm. Moreover, we investigate when the generic Karhunen-Loève expansion can be used to construct reproducing kernel Hilbert spaces (RKHSs) containing the paths of a version of the process. We further illustrate how the general theory can be applied, even in the absence of an explicitly known Karhunen-Loève expansion, by comparing the smoothness of the paths with the smoothness of the functions contained in the RKHS of the covariance function and by discussing some small ball probabilities. Key tools for our results are a recently shown generalization of Mercer’s theorem, spectral properties of the covariance integral operator, interpolation spaces of the real method, and compactness results for embeddings between classical function spaces.