We clarify the link between the notion of singular copula and the concept of support of the measure induced by a copula.

We present a proof of Sklar’s Theorem that uses topological arguments, namely compactness (under the weak topology) of the class of copulas and some density properties of the class of distribution functions.

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.

In this contribution we stress the importance of Sklar’s theorem and present a proof of this result that is based on the compactness of the class of copulas (proved via elementary arguments) and the use of mollifiers. More

We introduce a set of axioms for measures of non--exchan-geabi-lity for bivariate vectors of continuous and identically distributed random variables and give some examples together with possible applications in statistical models based on the copula function.

We present a general view of patchwork constructions of copulas that encompasses previous approaches based on similar ideas (ordinal sums, gluing methods, piecing-together, etc.). Practical applications of the new methodology are connected with the determination of copulas having specified behaviour in the tails, such as upper comonotonic copulas.

The book explores the state of the art on copulas, After covering the esseentilas of copula theory, the book addresses the issue of modelling dependence among components of a random vector using copulas. It presents copulas from the point of view of meaure theory, compares methods for the approximation of copulas, and discusses the Markov product for 2-copulas. Selected families of copulas are examined that possess appealing featuers from both theoretical and applied viewpoints. The book concludes with in-depth discussions on two generalisations of copulas: quasi- and semi-copulas.

Sklar’s theorem establishes the connection between a joint d-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∞ norm) of the class of copulas.