A1135
Title: Extremal negative dependence and the strongly Rayleigh property-part A
Authors: Alessandro Mutti - Politecnico di Torino (Italy)
Helene Cossette - Laval University (Canada)
Etienne Marceau - Laval University (Canada)
Patrizia Semeraro - Politecnico di Torino (Italy) [presenting]
Abstract: Negative dependence properties of multidimensional Bernoulli distributions are important in many areas of probability. However, the theory of negative dependence is more challenging than that of positive dependence, and there are still open problems. While extremal positive dependence is an important concept clearly defined by the notion of comonotonicity, extremal negative dependence is still an open issue. Extremal negative dependence is characterized in Frechet classes of multidimensional Bernoulli distributions by completing the theory of negative dependence with the theory of dependence orders. Specifically, we prove that in the class of multidimensional Bernoulli distributions, extremal negative dependence is properly represented by the notion of sigmacountermonotonicity. Indeed, building on the geometrical representation of the class of Bernoulli variables, it is proven that the class of sigmacountermonotonic distributions is a convex polytope in the simplex of multidimensional Bernoulli distributions, and it is proven that it is an antichain that satisfies some minimality conditions with respect to the strongest negative dependence orders.
