A joint mix is a random vector with a constant component-wise sum. The dependence structure of a joint mix minimizes some common objectives such as the variance of the component-wise sum, and it is regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and popular notions of negative dependence in statistics, such as negative correlation dependence, negative orthant dependence and negative association. A joint mix is not always negatively dependent in any of the above senses, but some natural classes of joint mixes are. We derive various necessary and sufficient conditions for a joint mix to be negatively dependent, and study the compatibility of these notions. For identical marginal distributions, we show that a negatively dependent joint mix solves a multi-marginal optimal transport problem for quadratic cost under a novel setting of uncertainty. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between negative dependence and the joint mix structure.

翻译：联合混合是指各分量之和恒为常数的随机向量。联合混合的相依结构能使分量之和的方差等常见目标函数最小化，因此被视为极端负相依概念。本文探讨了联合混合结构与统计学中流行的负相依概念（如负相关相依、负象限相依和负关联）之间的联系。联合混合并不总是满足上述任何意义上的负相依性，但其某些自然类别具有该性质。我们推导出联合混合具有负相依性的若干充要条件，并研究了这些概念的相容性。对于相同边际分布的情形，我们证明在一种新颖的不确定性设置下，具有负相依性的联合混合能解决二次成本的多边际最优传输问题。对该问题在异质边际分布下的分析揭示了负相依性与联合混合结构之间的权衡关系。