Necessary and sufficient conditions for realizability of point processes

TL;DR

This paper establishes exact conditions under which given functions can represent the density and pair correlations of a point process in various topological spaces, extending classical moment problems to infinite dimensions.

Contribution

It provides necessary and sufficient criteria for realizability of point processes with specified correlations, including cases with unbounded points and higher moments.

Findings

Derived conditions for point process realizability in various spaces

Extended classical moment problem to infinite-dimensional settings

Addressed cases with unbounded point configurations and higher moments

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.