  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.
