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It is only since the invention of Hallin's center-outward distribution function that optimal transport plans between probability measures have been recognized as multivariate analogues of univariate cumulative distribution and quantile functions. The potential for the construction of distribution-free hypothesis tests out of multivariate "ranks" and "quantiles" is enormous.
If the measures involved need to be inferred from data, the optimal transport plan needs to be estimated. A fundamental question underlying inference based upon estimated plans is their uniform consistency.
We provide a general theory delivering uniform consistency of random optimal transport plans between estimated distributions. The data generating process does not matter; the theory applies to time series, for instance. Besides absolute continuity, nothing is required of the true distributions, neither in terms of density nor support. The theory is based on the identification of an estimated optimal transport plan with a random closed set in an appropriate Fell space together with maximal cyclic monotonicity.
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