# Negatively associated point processes

A point process $\Phi$ is said to be negatively associated if for any finitely many bounded Borel subsets $B_1,B_2,...,B_n,$ we have that

$$\operatorname{Cov} \left( f\left(\Phi\left(B_1\right),\ldots,\Phi\left(B_l\right)\right)g\left(\Phi\left(B_{l+1}\right),\ldots,\Phi\left(B_{n}\right)\right)\right) \leq 0$$

whenever $f,g$ are non-decreasing, non-negative functions and $(B_1 \cup ... B_l) \cap (B_{l+1} \cup ... B_n) = \emptyset.$ Here $\Phi(B_i)$ denotes the random number of points within the set $B_i$.

This is an extension of the notion of positive association for random measures. There are many examples of positively associated point processes. A detailed study of Negative association by R. Pemantle in 2000 led to further interest in the concept of negative dependence. There are many examples of such measures in the discrete setting but i am not aware of any such example of spatial point processes i.e, point processes in $R^d$.

In a paper of R. Lyons, it is proved that discrete determinantal probability measures are negatively associated. Though i expect it to work for spatial determinantal processes as well, currently it stands unproved.

Is someone aware of any negatively associated point process ? Or do you have suggestion for any other point process that you suspect will be negatively associated ?

• On a bounded region, isn't the process of a fixed number of independent points negatively associated? – Ori Gurel-Gurevich Nov 25 '11 at 0:11
• Thanks Ori. I am puzzled as to why i missed this earlier. In fact, one might be able to show that superposition of independent NA point processes will lead to a NA point process. So, a point process with $n$ uniformly distributed points in each cube of the $Z^d$ lattice will be negatively associated point process on the entire plane. – yogesh Nov 25 '11 at 16:59