Symmetric Bernoulli distributions and minimal dependence copulas

The key result of this paper is to characterize all multivariate symmetric Bernoulli distributions whose sum is minimal under the convex order. In doing so, we automatically characterize extremal negative dependence among Bernoulli variables, since multivariate distributions with minimal convex sums are known to be strongly negative dependent. Moreover, beyond its interest per se, this result provides insight into negative dependence within the class of copulas. In particular, two classes of copulas can be built from multivariate symmetric Bernoulli distributions: extremal mixture copulas and FGM copulas. We analyze the extremal negative dependence structures of copulas constructed from symmetric Bernoulli vectors with minimal convex sums and explicitly find a class of minimal dependence copulas. This analysis is completed by investigating minimal pairwise dependence measures and correlations. Our main results derive from the geometric and algebraic representations of multivariate symmetric Bernoulli distributions, which effectively encode key statistical properties.