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We establish a Karhunen-Lo`eve expansion for generic centered, second orderstochastic processes, which does not rely on topological assumptions. Wefurther investigate in which norms the expansion converges and derive exactaverage rates of convergence for these norms. For Gaussian processes weadditionally prove certain sharpness results in terms of the norm. Moreover, weshow that the generic Karhunen-Lo`eve expansion can in some situations be usedto construct reproducing kernel Hilbert spaces (RKHSs) containing the paths ofa version of the process. As applications of the general theory, we compare thesmoothness of the paths with the smoothness of the functions contained in theRKHS of the covariance function and discuss some small ball probabilities. Keytools for our results are a recently shown generalization of Mercer's theorem,spectral properties of the covariance integral operator, interpolation spacesof the real method, and for the smoothness results, entropy numbers ofembeddings between classical function spaces.

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