Expansions for Gaussian processes and Parseval frames

We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived. In the end an extension of this result to Gaussian stationary processes with convex covariance function is established.