A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based on the identification of such a mapping with a closed subset of a Cartesian product of Euclidean spaces and leveraging tools from random set theory. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically yields local uniform convergence of the associated mappings. Viewing such mappings as optimal transport plans between probability measures with respect to the squared Euclidean distance as cost function yields consistency results for notions of multivariate ranks and quantiles based on optimal transport, notably the empirical center-outward distribution and quantile functions.

中文翻译：

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估计的最佳运输计划的图形和统一一致性

提供了一种通用理论，该理论提供了最大循环单调映射的收敛性，该映射包含对欧几里得空间上可能的随机概率测度对的序列的耦合测度的支持。该理论基于使用欧几里得空间的笛卡尔积的封闭子集识别这种映射，并利用随机集理论中的工具。适当的 Fell 空间中的弱收敛与最大循环单调性一起自动产生相关映射的局部均匀收敛。将此类映射视为关于平方欧几里德距离的概率度量之间的最佳传输计划作为成本函数，可以为基于最佳传输的多元等级和分位数的概念产生一致性结果，