On metric spaces of subcopulas.

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. [ABSTRACT FROM AUTHOR]