The authors study 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 density and pair correlation function of some point process in a topological space \(X\). In this case, \(\rho_1\) and \(\rho_2\) are said to be realizable. This is an infinite-dimensional version of the classical truncated moments problem, which is solved by reducing it to checking the positivity of some linear functionals and by a subsequent application of the extension theorems for positive functionals. This approach yields very general necessary conditions, which in special cases yield necessary conditions known from the physics literature. The use of the positive extension techniques is a very stimulating and innovative approach in stochastic geometry.
In order to interpret the extended functional as the Lebesgue integral (and thus to come up with sufficient conditions), the authors restrict their attention to processes with hard-core exclusion or point processes with finite third-order moment for the number of points in a compact set. In these cases, the conditions become necessary and sufficient.
The authors also characterise the existence of stationary point processes in the case that the function \(\rho_1\) is constant and \(\rho_2\) depends on the difference of its arguments.