Abstract

Copulas have emerged over the last decades as primary statistical tools for modelling dependence between random variables. A copula is classically understood as a cumulative distribution function on the unit hypercube with standard uniform margins – we refer to such distributions as “Sklar’s copulas”, owing to their central role in the decomposition of multivariate distributions established by the celebrated Sklar’s theorem. A standard argument in favour of copula models is that they separate the dependence structure (encoded by the copula) from the marginal behaviour of individual components. However, this interpretation holds only in the continuous case: outside it, copulas lose their “margin-free” nature, rendering Sklar’s construction unsuitable for modelling dependence between non-continuous variables. In this work, we argue that the notion of a copula need not be confined to Sklar’s framework. We propose an alternative definition—universal copulas—based on a more precise characterisation of dependence. This new definition agrees with Sklar’s copulas in the continuous case, but yields distinct and more suitable constructions in discrete or mixed settings. Universal copulas retain key properties such as margin-freeness, making them sound and effective beyond the continuous realm. We illustrate their use through examples involving discrete variables and mixed pairs, such as one continuous and one Bernoulli variable.