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

TL;DR

This paper establishes that association properties of finite-dimensional marginals imply the same for infinite sequences and random measures in Polish spaces, simplifying proofs and extending previous results.

Contribution

It proves that association in finite marginals implies association in infinite sequences and measures in Polish spaces, avoiding previous restrictive assumptions.

Findings

Association of finite marginals implies infinite sequence association.

Negative association of finite marginals implies negative association of random measures.

Negative association is preserved under weak convergence of random measures.

Abstract

In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space (POP), 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 [Lindqvist 1988]. We use these results to show on Polish 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 [Poinas et al. 2017] and [Lyons 2014] which restrict to point processes in Euclidean spaces and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.