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On the stochastic convergence of representations based on Wasserstein metrics. (English) Zbl 0770.60012

Under some regularity conditions an optimal coupling of two probability measures on a Hilbert space can be found in the form \((X,H(X))\) for some monotone function \(H\). It is shown that convergence of \(P_ n\) to \(P\) in distribution implies a.s. convergence of the optimal coupling functions \(H_ n\) in the finite-dimensional case. For the infinite-dimensional case a counterexample is given.

MSC:

60E05 Probability distributions: general theory
60B10 Convergence of probability measures
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