Necessary and sufficient conditions for realizability of point processes

We giüe necessary and sufficient conditions for a pair of (generalized) functions ρ1 (r1) and ρ2 (r1, r1), 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 üersion 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. © 2011 Institute of Mathematical Statistics.