Algorithmic need for subcopulas

Abstract

One of the efficient ways to describe the dependence between random variables is by describing the corresponding copula. For continuous distributions, the copula is uniquely determined by the corresponding distribution. However, when the distributions are not continuous, the copula is no longer unique, what is unique is a subcopula, a function C(u, v) that has values only for some pairs (u, v). From the purely mathematical viewpoint, it may seem like subcopulas are not needed, since every subcopula can be extended to a copula. In this paper, we prove, however, that from the algorithmic viewpoint, it is, in general, not possible to always generate a copula. Thus, from the algorithmic viewpoint, subcopulas are needed.