An analysis of the R\"uschendorf transform - with a view towards Sklar's Theorem

Abstract

In many applications including financial risk measurement, copulas have shown to be a powerful building block to reflect multivariate dependence between several random variables including the mapping of tail dependencies. A famous key result in this field is Sklar's Theorem. Meanwhile, there exist several approaches to prove Sklar's Theorem in its full generality. An elegant probabilistic proof was provided by L. R\"{u}schendorf. To this end he implemented a certain "distributional transform" which naturally transforms an arbitrary distribution function $F$ to a flexible parameter-dependent function which exhibits exactly the same jump size as $F$. By using some real analysis and measure theory only (without involving the use of a given probability measure) we expand into the underlying rich structure of the distributional transform. Based on derived results from this analysis (such as Proposition 2.5 and Theorem 2.12) including a strong and frequent use of the right quantile function, we revisit R\"{u}schendorf's proof of Sklar's theorem and provide some supplementing observations including a further characterisation of distribution functions (Remark 2.3) and a strict mathematical description of their "flat pieces" (Corollary 2.8 and Remark 2.9).