Convergence Types and Rates in Generic Karhunen-Lo\`eve Expansions with Applications to Sample Path Properties

Abstract

We establish a Karhunen-Lo`eve 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 we additionally prove certain sharpness results in terms of the norm. Moreover, we show that the generic Karhunen-Lo`eve expansion can in some situations be used to construct reproducing kernel Hilbert spaces (RKHSs) containing the paths of a version of the process. As applications of the general theory, we compare the smoothness of the paths with the smoothness of the functions contained in the RKHS of the covariance function and discuss 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 for the smoothness results, entropy numbers of embeddings between classical function spaces.