Copulas with continuous, strictly increasing singular conditional distribution functions

Using Iterated Function Systems induced by so-called modifiable transformation matrices T   and tools from Symbolic Dynamical Systems we first construct mutually singular copulas AT⋆ with identical (possibly fractal or full) support that are at the same time singular with respect to the Lebesgue measure λ2λ2 on [0,1]2[0,1]2. Afterwards the established results are utilized for a simple proof of the existence of singular copulas AT⋆ with full support for which all conditional distribution functions y↦FxAT⋆(y) are continuous, strictly increasing and have derivative zero λ  -almost everywhere. This result underlines the fact that conditional distribution functions of copulas may exhibit surprisingly irregular analytic behavior. Finally, we extend the notion of empirical copula to the case of non-i.i.d. data and prove uniform convergence of the empirical copula En′ corresponding to almost all orbits of a Markov process usually referred to as chaos game to the singular copula AT⋆. Several examples and graphics illustrate both the chosen approach and the main results.