Necessary and sufficient conditions for realizability of point processes

Abstract

We give necessary and sufficient conditions for a pair of (generalized) functions ρ1(r1) and ρ2(r1, r2), ri ∈ X, to be the density and pair correlations of some point process in a topological space X, for example, ℝd, ℤd 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.