  This work studies finite sample approximations of the exact and entropic
regularized Wasserstein distances between centered Gaussian processes and, more
generally, covariance operators of functional random processes. We first show
that these distances/divergences are fully represented by reproducing kernel
Hilbert space (RKHS) covariance and cross-covariance operators associated with
the corresponding covariance functions. Using this representation, we show that
the Sinkhorn divergence between two centered Gaussian processes can be
consistently and efficiently estimated from the divergence between their
corresponding normalized finite-dimensional covariance matrices, or
alternatively, their sample covariance operators. Consequently, this leads to a
consistent and efficient algorithm for estimating the Sinkhorn divergence from
finite samples generated by the two processes. For a fixed regularization
parameter, the convergence rates are {\it dimension-independent} and of the
same order as those for the Hilbert-Schmidt distance. If at least one of the
RKHS is finite-dimensional, we obtain a {\it dimension-dependent} sample
complexity for the exact Wasserstein distance between the Gaussian processes.
