Necessary and sufficient conditions for realizability of point processes

Abstract

We give necessary and sufficient conditions for a pair of (generalized) functions $\rho_1(\mathbf{r}_1)$ and $\rho_2(\mathbf{r}_1,\mathbf{r}_2)$, $\mathbf{r}_i\in X$, to be the density and pair correlations of some point process in a topological space $X$, for example, $\mathbb {R}^d$, $\mathbb {Z}^d$ or a subset of these. This is an infinite-dimensional version of the classical "truncated moment" problem. Standard techniques apply in the case in which there can be only a bounded number of points in any compact subset of $X$. Without this restriction we obtain, for compact $X$, strengthened conditions which are necessary and sufficient for the existence of a process satisfying a further requirement---the existence of a finite third order moment. We generalize the latter conditions in two distinct ways when $X$ is not compact.