Measure-Preserving Functions and the Independence Copula

Abstract

We solve a problem recently proposed by Kolesárová et al. Specifically, we prove that a necessary and sufficient condition for a given copula to be the independence or product copula is for the pair of measure-preserving transformations representing the copula to be independent as random variables.We provide examples of such pairs for the well-known Cantor, Peano, and Hilbert curves. Moreover, a general constructive method is given for the representation of copulae in terms of measure-preserving transformations. In particular, we apply numbers representation systems to the study of self-similar copulae properties.Mathematics Subject Classification (2010). Primary 60E05; Secondary 28D05.

Singular Functions with Applications to Fractal Dimensions and Generalized Takagi Functions

“…We introduce a number representation system which generalizes another system of representation B used by the authors in [2]. The new system is also denoted by B, and it is defined as follows:…”

Shuffles of copulas and a new measure of dependence

“…Proof. We shall prove only (7) which shows that µ * ν is a doubly stochastic measure when the measures µ and ν are doubly stochastic and inducible by copulas. Let A and B be copulas and…”

Essential closures and supports of multivariate copulas

“…One such case is the case of bivariate complete dependence copulas. And if we assume that the random variables are uniform on [0, 1], there is a measure-preserving transformation on [0, 1] connecting the two random variables. It has been observed that the graph of such a function and the support of the corresponding copula are closely related.…”

“…It has been observed that the graph of such a function and the support of the corresponding copula are closely related. For instance, it has been shown in [2] that the mass of a copula is concentrated on the graph of a corresponding function (V C (gr f ) = 1). In our work, we obtain that the support of such copula is an essential closure of the graph of a "refinement"of the function.…”

“…which is the main diagonal of the unit cube [0, 1] 3 . Similarly, supp M n is the main diagonal of the hypercube [0, 1] n .…”