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A joint mix is a random vector with a constant component-wise sum. Thedependence structure of a joint mix minimizes some common objectives such asthe variance of the component-wise sum, and it is regarded as a concept ofextremal negative dependence. In this paper, we explore the connection betweenthe joint mix structure and popular notions of negative dependence instatistics, such as negative correlation dependence, negative orthantdependence and negative association. A joint mix is not always negativelydependent in any of the above senses, but some natural classes of joint mixesare. We derive various necessary and sufficient conditions for a joint mix tobe negatively dependent, and study the compatibility of these notions. Foridentical marginal distributions, we show that a negatively dependent joint mixsolves a multi-marginal optimal transport problem for quadratic cost under anovel setting of uncertainty. Analysis of this optimal transport problem withheterogeneous marginals reveals a trade-off between negative dependence and thejoint mix structure.

Joint mixability and notions of negative dependence