Some remarks on associated random fields, random measures and point processes

Abstract

In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space with a closed partial order (POP space), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in [38]. We use these results to show on POP spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in [47] and [40] which restrict to point processes in R d and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.

Percolation of Repulsive Particles on Graphs

“…One can take advantage of known examples of negatively associated sequences (see e.g. [7]). In particular one can use negatively associated Gaussian sequences.…”