Summary: Let \(X\) be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators \(a(\Delta )\) indexed by all measurable, relatively compact sets \(\Delta\) in \(X\) (a quantum stochastic process over \(X\)). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over \(X\) having the same correlation measure. Furthermore, the operators \(a(\Delta )\) can be realized as multiplication operators in the \(L^2\)-space with respect to this point process. In the proof, we utilize the notion of \(\star\)-positive definiteness, proposed in [Yu. G. Kondratiev and T. Kuna, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, No. 2, 201–233 (2002; Zbl 1134.82308)]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space \(X\), which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way.