M. Díaz Carrillo

Characterization of all copulas associated with non-continuous random variables

We introduce a constructive method, by means of a doubly stochastic measure, to describe all the copulas that, in view of Sklars Theorem, are able to connect a bivariate distribution to its marginals. We use this to give the lower and upper optimal bounds for all the copulas that extend a given subcopula.

On the duality of aggregation operators and k-negations

In this paper we study a class of duality functions given by the solution of a system of functional equations related to the De Rham system. With the aid of a generalized dyadic representation system in the unit interval, we study a negation N which is a duality function for pairs of operators satisfying certain boundary conditions. New properties of N are investigated, including its singularity and fractal dimensions for several related sets. As an application we obtain an explicit expression for k-negations.

An approximate functional Radon-Nikodym theorem

AbstractWe prove that for two linear and positive functionals (not necessarily Daniell)J andI on a lattice unitary algebraB of functions such thatJ is absolutely continuous with respect toI, one can expressJ as follows:

On Concordance Measures and Copulas with Fractal Support

J(f) = \mathop {\lim }\limits_m I(fv_m )

Fubini-integral metrics

, where (vm)m is a fixed sequence inB, for allf inB. This result is the “functional” similar of a previous deep result due to C. Fefferman.The comments and the counterexamples which we are introducing show that the main result (i.e sequential approximation) cannot be improved.

FINITELY ADDITIVE INTEGRATION AND LOCAL INTEGRATION WITH RESPECT TO UPPER INTEGRALS

In [ 3 ] for general integral metric q an integral extension of Lebesgue power was discussed. In this paper we introduce the abstract Daniell-Loomis spaces R p , p real, 0 p q -measurable functions with finite “ p -norm”, and study their basic properties.

The Hausdorff dimension of the level sets of Takagi’s function

Copulas can be used to describe multivariate dependence structures. We explore the role of copulas with fractal support in the study of association measures.

Copulas and associated fractal sets

In this paper we start from previous results obtained in [7] on the abstract space of Daniell-Loomis integrable functionsL, which is constructed like to the Daniell extension process, but without continuity assumptions on the elementary integral.The localized integral is used to prove thatL consists of those functions whose local upper and lower integrals are equal and finite, or thatL is closed with respect to improper integration.Our results are also holded in integration with respect to finitely additive measures.