The Annals of Probability

Notes on the Wasserstein Metric in Hilbert Spaces

Juan Antonio Cuesta and Carlos Matran

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Abstract

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.

Article information

Source
Ann. Probab., Volume 17, Number 3 (1989), 1264-1276.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991269

Digital Object Identifier
doi:10.1214/aop/1176991269

Mathematical Reviews number (MathSciNet)
MR1009457

Zentralblatt MATH identifier
0688.60011

JSTOR
links.jstor.org

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Wasserstein distance representation theorem Hilbert spaces central limit theorem

Citation

Cuesta, Juan Antonio; Matran, Carlos. Notes on the Wasserstein Metric in Hilbert Spaces. Ann. Probab. 17 (1989), no. 3, 1264--1276. doi:10.1214/aop/1176991269. https://projecteuclid.org/euclid.aop/1176991269


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