  We provide upper bounds of the expected Wasserstein distance between a
probability measure and its empirical version, generalizing recent results for
finite dimensional Euclidean spaces and bounded functional spaces. Such a
generalization can cover Euclidean spaces with large dimensionality, with the
optimal dependence on the dimensionality. Our method also covers the important
case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper
bounds for functional data distributions whose coordinates decay geometrically
or polynomially. Moreover, our bounds of the expected value can be combined
with mean-concentration results to yield improved exponential tail probability
bounds for the Wasserstein error of empirical measures under Bernstein-type or
log Sobolev-type conditions.
