Inclusion-exclusion and point processes in Polish spaces
\textit{R. V. Ambartzumyan} and \textit{G. S. Sukiasyan} have considered [Acta Appl. Math. 22, No. 1, 15-31 (1991; Zbl 0723.60051)] the problem of constructing point processes in the Euclidean space \(R^d\), starting from the so-called absolute densities (equivalently, the correlation functions). A criterion based on a kind of combinatorial inclusion-exclusion principle was derived, telling when a given system of functions happens to be the system of absolute densities of a point process in \(R^d\). This criterion was applied in the paper quoted above to the class of functions of the form \(f(x_1,\dots,x_n) =\alpha^n\prod_n h(x_i,x_j)\), where \(\alpha\) is a positive constant, \(h(x_i, x_j): R^d \times R^d\mapsto [0,1]\) is a symmetrical function, the product is extended over all unordered pairs \(\{i,j\} \subseteq \{1,\dots,n\}\). A sufficient condition was obtained in the paper quoted above, under which the functions \(f(x_1,\dots,x_n)\) become absolute densities for some point process. Point processes with density functions of such special form play an important role [see \textit{R. V. Ambartzumyan}, ibid. 22, No. 1, 3-14 (1991; Zbl 0726.60046) and in: New trends in probability and statistics, Vol. 1, 655-667 (1991; Zbl 0787.60056)] in the theory of classical Gibbs point processes. The present paper considers similar problems for point processes in general Polish space. The author formulates an inclusion-exclusion criterion within that general setting with no assumption of existence of densities and works directly with ``reduced'' moment measures. This permits in particular to extend the results of the first quoted paper to point processes on lattices and marked point processes in \(R^d\).
scientific article; zbMATH DE number 1193058