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

In [32] we presented an overview of concepts, facts and results on triangle functions based on the notions of t-norm, copula, (generalized) convolution, semicopula, quasi-copula. Here, we continue our presentation. In particular, we treat the concept of duality and study a few important cases of functional equations and inequalities for triangle functions like, e.g., convolution, Cauchy's equation, dominance, and Jensen convexity.

It was shown [8] that uniform boundedness in a Serstnev PN space $(V,nu,tau,tau^*)$, (named boundedness in the present setting) of a subset $Asubset V$ with respect to the strong topology is equivalent to the fact that the probabilistic radius $R_A$ of $A$ is an element of $D^+$. Here we extend the equivalence just mentioned to a larger class of PN spaces, namely those PN spaces that are topological vector spaces (briefly TV spaces), but are not Serstnev PN spaces. Section 2 presents a characterization of those PN spaces, whether they are TV spaces or not, in which the equivalence holds. In Section 3, a characterization of the Archimedeanity of triangle functions $tau^*$ of the type $tau_{T,L}$ is given. This work is a partial solution to a problem of comparing the concepts of distributional boundedness ($D$--bounded in short) and that of boundedness in the sense of associated strong topology.

We survey the measures of association that are based on bivariate copulas. Almost no proof will be reported, although an exception is made in the case of the Schweizer--Wolff measure, since the details of the proof are mainly contained in Wolff's Ph.D. dissertation, which is not readily available.

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.

It presents the main features of the field of copulas

The copulae of a few stochastic processes related to the Brownian motion are derived; specifically, if $(X_t)$ is one such process, the copula of the pair $(X_s,X_t)$ is determined for $s<t$.

This paper starts with a few critical considerations about the use of copulas in applications, mainly in the field of Mathematical Finance. Two points will be stressed: (i) the construction of asymmetric copulas and (ii) the construction of multivariate copulas. Also, it briefly touches on the long-standing problem of compatibility.

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.

We present a way of consodering a stochastic process ${B_t:tge 0} with values in $mathbb{R}^2$ such each component is a Brownian motion. The distribution function of $B_t$, for each $t$, is obtained as the copula of the distribution functions of the components. In nthis way "coupled Brownian motion" is obtained.The (one-dimensional) Brownian motion is the example of a stochastic process that (a) is a Markov process, (b) is a martingale in continuous time, and (c) is a Gaussian process. It will be seen that while the coupled Brownian motion is still a Markov process and a martingale, it is not in general a Gaussian process.

Two algebraic notions, power of an associative binary function and nilpotency, are used in order to show that every bivariate Archimedean copula $C$ is isomorphic to either the independence copula $Pi_2$, if it is strict, or to the lower Fr'{e}chet--Hoeffding bound $W_2$, if it is nilpotent.

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

Idempotent copulae have been characterised, in an implicit form, in cite{Sem02}; here we look at a few well known classes of copulas, namely, Fréchet copulas, ordinal sums, Archimedean copulas, copulas of the type $C(u,v)=uvpm f(u),g(v)$ and a special subset of copulas represented through measure-preserving transformations, and characterise those among these classes that are idempotent.

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.

We prove that the ordinal sum of n-copulas is always an n-copula and show that every copula may be represented as an ordinal sum, once the set of its idempotents is known. In particular, it will be shown that every copula can be expressed as the ordinal sum of copulas having only trivial idempotents. As a by-product, we also characterize all associative copulas whose n-ary forms are n-copulas for all n.

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.