Sklar's Theorem revisited: an elaboration of the Rüschendorf transform approach

Frank Oertel

Published 2013-12-17, updated 2015-01-21Version 2

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 non-trivial results from topology and functional analysis. An elegant and mostly probabilistic proof was provided by L. R\"uschendorf in 2009. We will revisit R\"uschendorf's - very short - 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 R\"uschendorf's proof mainly splits into two parts: a purely analytic one (without any assumption on randomness) and a probabilistic one. To this end, we slightly generalise R\"uschendorf's approach from 2009, allowing us to derive a result which might become very useful regarding further applications of copula theory.