Positive Dependence Properties of Point Processes

Abstract

There are many ways of describing positive dependence, for example the strong FKG inequalities and association. It is known that for Bernoulli random variables the strong FKG inequalities are equivalent to all the conditional distributions being associated, which is in turn equivalent to all the conditional distributions having positively correlated marginals. These and similar definitions are extended to point processes on $\mathbb{R}^d$. Examples are given to show that, unlike the analogous Bernoulli random variable case, these conditions are no longer equivalent, although some are implied by others.