Sklar’s theorem obtained via regularization techniques

Sklar’s theorem establishes the connection between a joint dd-dimensional distribution function and its univariate marginals. Its proof is straightforward when all the marginals are continuous. The hard part is the extension to the case where at least one of the marginals has a discrete component. We present a new proof of this extension based on some analytical regularization techniques (i.e., mollifiers) and on the compactness (with respect to the L∞L∞ norm) of the class of copulas.