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Notes on the Wasserstein Metric in Hilbert Spaces

1989; Institute of Mathematical Statistics; Volume: 17; Issue: 3 Linguagem: Inglês

10.1214/aop/1176991269

2168-894X

Juan A. Cuesta‐Albertos, Carlos Matrán,

Advanced Banach Space Theory

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.