Abstract In this work, we show that two distance functions independently defined on the space of subcopulas are topological equivalent. In this process, we also defined another distance function equivalent to the first two distances. Moreover, all metric spaces of subcopulas with fixed domain under the supremum distance are metric subspaces under this new distance function. We also prove that the Sklar correspondence can be viewed as a Lipschitz map under these metrics. Thus, the rate of convergence of empirical subcopulas can be computed directly from the rate of convergence of empirical distributions. The same holds for other statistics results.

中文翻译：

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关于子联结的度量空间

摘要 在这项工作中，我们证明了在 subcopula 空间上独立定义的两个距离函数是拓扑等价的。在这个过程中，我们还定义了另一个等价于前两个距离的距离函数。而且，在最高距离下的所有具有固定域的子系的度量空间都是这个新距离函数下的度量子空间。我们还证明了 Sklar 对应可以被视为在这些指标下的 Lipschitz 图。因此，经验 subcopulas 的收敛速度可以直接从经验分布的收敛速度计算。这同样适用于其他统计结果。