Sklar's Theorem Revisited: An Elaboration of the Rüschendorf Transform Approach

Abstract

In many applications including financial risk measurement a certain class of multivariate distribution functions, copulas, has shown to be a powerful building block to reflect multivariate dependence between several random variables including the mapping of tail dependencies.A key result in this field is Sklar's Theorem which roughly states that any n-variate distribution function can be written as a composition of a suitable copula and an n-dimensional vector whose components are given by univariate marginal distribution functions, and that conversely the composition of an arbitrary copula and an arbitrary n-dimensional vector, consisting of n one-dimensional distribution functions (which need not be continuous), again is a n-variate distribution function whose i-th marginal is precisely the i-th component of the n-dimensional vector of the given distribution functions.Meanwhile, in addition to the original sketch of a proof by Sklar himself, there exist several approaches to prove Sklar's Theorem in its full generality, mostly under inclusion of probability theory and mathematical statistics but recently also rather technically under inclusion of nontrivial results from topology and functional analysis. An elegant probabilistic sketch of a proof was provided by L. Ruschendorf.We will revisit Ruschendorf's - very short (and seemingly incomplete) - proof and elaborate important details to lighten the understanding of the basic underlying ideas of this proof including the major role of the so called "distributional transform". Thereby, we will recognise that Ruschendorf's proof mainly splits into two parts: a purely real-analytic one (without any assumption on randomness) and a probabilistic one.To this end, we slightly generalise Ruschendorf's approach, allowing us to derive Theorem 2.9 and Lemma 2.13 - two results which might become very useful, in particular with respect to a simulation of random variables. To this end, we also provide a strict mathematical description of "flat pieces" of a right-continuous and non-decreasing real-valued function, leading to Corollary 2.6.