Notes on the Wasserstein Metric in Hilbert Spaces

Let $(X, Y)$ be a pair of Hilbert-valued random variables for which the Wasserstein distance between the marginal distributions is reached. We prove that the mapping ω → (X(ω), Y(ω)) is increasing in a certain sense. Moreover, if Y satisfies a nondegeneration condition, we can take $X = T(Y)$ with T monotone in the sense of Zarantarello. We apply these results to obtain a proof of the central limit theorem (CLT) in Hilbert spaces which does not make use of the CLT for real-valued random variables.