The Topology of Negatively Associated Distributions

Abstract

We consider the sets of negatively associated (NA) and negatively correlated (NC) distributions as subsets of the space $\mathcal{M}$ of all probability distributions on $\mathbb{R}^n$, in terms of their relative topological structures within the topological space of all measures on a given measurable space. We prove that the class of NA distributions has a non-empty interior with respect to the topology of the total variation metric on $\mathcal{M}$. We show however that this is not the case in the weak topology (i.e. the topology of convergence in distribution), unless the underlying probability space is finite. We consider both the convexity and the connectedness of these classes of probability measures, and also consider the two classes on their (widely studied) restrictions to the Boolean cube in $\mathbb{R}^n$.