Proving something that seems intuitively obvious does not make for the most glamorous of papers, but that does not detract from the value of such a contribution to our knowledge. The author shows that the pointwise convergence of a sequence of copulas to a [given] copula is “equivalent to the convergence of the corresponding endographs.” In almost all cases (with the exception of at most countably many), the above is also equivalent to the convergence of the upper or lower levels.

A brief introduction defines the meaning of an endograph and the upper- and lower-level sets. This is followed by a section on notation and some preliminary definitions. The third section deals with the properties of the upper and lower functions of a copula, first showing that the upper- and lower-level functions will have the same discontinuities, and then defining some properties for a radius-vector function. The radius-vector function is then used to show that there can only be countably many discontinuities.

The proofs in section 3 are concretized using an example copula. The example constructs the sequence of copulas with fractal support based on iterated function systems. The subsequent section presents the primary result of the paper, the proof that pointwise convergence of the sequence of copulas does in fact imply the convergence of the upper and lower endographs.