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为常数的随机向量。这种依赖结构最小化了某些常见目标（如分量之和的方差），被视为极端负相关的概念。本文探讨了联合混合结构与统计学中流行的负相关概念（如负相关依赖、负象限依赖和负关联）之间的联系。联合混合并不总是满足上述任何一种负相关意义，但某些自然类别的联合混合满足该性质。我们推导了联合混合具有负相关性的各种必要充分条件，并研究了这些概念的兼容性。对于相同的边际分布，我们证明，在一种新颖的不确定性设定下，具有负相关性的联合混合可解决二次成本的多边际最优传输问题。对具有异质边际分布的该最优传输问题的分析揭示了负相关性与联合混合结构之间的权衡。