Necessary and sufficient conditions for realizability of point processes

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.