Open Access
July, 1989 Notes on the Wasserstein Metric in Hilbert Spaces
Juan Antonio Cuesta, Carlos Matran
Ann. Probab. 17(3): 1264-1276 (July, 1989). DOI: 10.1214/aop/1176991269


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.


Download Citation

Juan Antonio Cuesta. Carlos Matran. "Notes on the Wasserstein Metric in Hilbert Spaces." Ann. Probab. 17 (3) 1264 - 1276, July, 1989.


Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0688.60011
MathSciNet: MR1009457
Digital Object Identifier: 10.1214/aop/1176991269

Primary: 60E05
Secondary: 60B12

Keywords: central limit theorem , ‎Hilbert spaces , representation theorem , Wasserstein distance

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
Back to Top