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 $\omega \rightarrow (X(\omega), Y(\omega))$ 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.