The key result of this paper is to find all the joint distributions of random vectors whose sums $S=X_1+\ldots+X_d$ are minimal in convex order in the class of symmetric Bernoulli distributions. The minimal convex sums distributions are known to be strongly negatively dependent. Beyond their interest per se, these results enable us to explore negative dependence within the class of copulas. In fact, there are two classes of copulas that can be built from multivariate symmetric Bernoulli distributions: the extremal mixture copulas, and the FGM copulas. We study the extremal negative dependence structure of the copulas corresponding to symmetric Bernoulli vectors with minimal convex sums and we explicitly find a class of minimal dependence copulas. Our main results stem from the geometric and algebraic representations of multivariate symmetric Bernoulli distributions, which effectively encode several of their statistical properties.

翻译：本文的关键结果是在对称伯努利分布类中，找出使得其和 $S=X_1+\ldots+X_d$ 在凸序意义下最小的随机向量联合分布。已知具有最小凸序和的分布呈现强负依赖关系。除其本身的理论价值外，这些结果使我们能够探索连接函数类内部的负依赖结构。事实上，基于多元对称伯努利分布可构造两类连接函数：极值混合连接函数与FGM连接函数。我们研究了对应具有最小凸序和的对称伯努利向量的连接函数的极值负依赖结构，并显式地发现了一类最小依赖连接函数。本文主要结论源自多元对称伯努利分布的几何与代数表示，这些表示有效地编码了其若干统计特性。