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A class of Wasserstein metrics for probability distributions. (English) Zbl 0582.60002

Let (S,d) be a complete separable metric space and \({\mathfrak X}\) be the space of all S-valued random variables given on a probability space without atoms. Let \({\mathfrak M}_ p(p\geq 1)\) be the space of all Borel probability measures P on S with finite p-moment, i.e. \(\int_{S}d^ P(x,a)P(dx)<\infty\) for some \(a\in S\). The \(L^ P\)-Wasserstein distance is defined by \[ W_ P(P_ 1,P_ 2)=\inf \{{\mathfrak L}_ p(X,Y);\quad X,Y\in {\mathfrak X},\Pr_ X=P_ 1,\Pr_ Y=P_ 2\} \] where \(p\in [1,\infty]\) and \({\mathfrak L}_ p(X,Y)=\{{\mathbb{E}} d^ P(X,Y)\}^{1/p},\quad p<\infty,\) \({\mathfrak L}_{\infty}(X,Y)= \sup_{\Pr}d(X,Y).\) In §1 of the paper, the metrical and topological properties of (\({\mathfrak M}_ p,W_ p)\) are investigated.
As is well-known [see the reviewer’s survey, Teor. Veroyatn. Primen. 29, No.4, 625-653 (1984; Zbl 0565.60010); English translation in Theory Probab. Appl. 29, 647-676 (1985)] the main theoretical problems connected with \(W_ p\)-distance (as a particular case of the so-called ”marginal problems”) consist of dual and explicit descriptions of \(W_ p\). The main result of the paper (see §2) is an explicit representation of the \(L^ 2\)-Wasserstein distance \(W_ 2(P_ 1,P_ 2)\) for Gaussian measures \(P_ 1\) and \(P_ 2\) on \(R^ n\).
Reviewer: S.T.Rachev

MSC:

60A10 Probabilistic measure theory
60E05 Probability distributions: general theory

Citations:

Zbl 0565.60010
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