Inclusion-exclusion and point processes (Q756849)

For random point processes in the d-dimensional Euclidean space \({\mathbb{R}}^ d\) the notions of densities and relative densities are considered. For any bounded Borel set D in \({\mathbb{R}}^ d\), the relative density \(V_ D(x_ 1,...,x_ n)\) (of order n) at the n-tuple \((x_ 1,...,x_ n)\in {\mathbb{R}}^{nd}\) of locations in D is related to the probability that in each of the infinitesimal neighborhoods of the locations \(x_ 1,...,x_ n\) there is exactly one point of a random point configuration which has exactly n points in D, whereas the density \(f(x_ 1,...,x_ n)\) is related to the probability that in the neighborhoods of \(x_ 1,...,x_ n\) there is exactly one point of an arbitrary configuration. The problem is investigated under which conditions, for given systems of functions \(\{V_ D(x_ 1,...,x_ n)\}\) and \(\{f(x_ 1,...,x_ n)\}\), there exists a random point process such that \(\{V_ D(x_ 1,...,x_ n)\}\) and \(\{f(x_ 1,...,x_ n)\}\) are the relative densities and the densities of this point process, respectively. The approach used to solve this problem is combinatorial and based on a well-known inclusion-exclusion rule. It is shown how the existence conditions can be specified when the functions \(f(x_ 1,...,x_ n)\) admit a certain product form.