Joint Mixability and Notions of Negative Dependence

Abstract

A joint mix (JM) 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 these 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 multimarginal 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. Funding: T. Koike was supported by the Japan Society for the Promotion of Science [Grant JSPS KAKENHI JP21K13275]. R. Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03823 and RGPAS-2018-522590].