Some results on the convergence of (quasi-) copulas

It is shown that pointwise convergence of a sequence (An)n∈N(An)n∈N of copulas to a copula A   is equivalent (1) to the convergence of the corresponding endographs and (2) to the convergence of the corresponding upper (or lower) αα-levels for all but at most countably many αα in [0,1] (all with respect to the Hausdorff metric). Examples are given that show that the countably many exceptions in (2) cannot be omitted. It is furthermore shown that the main results also hold on the bigger class of quasi-copulas.

► Pointwise convergence of copulas is equivalent to convergence of their endographs.
► Convergence of copulas is equivalent to convergence of almost all upper level sets.
► Convergence of copulas is equivalent to convergence of almost all lower level sets.
► The convergence results also hold for the class of quasi-copulas.