Some remarks on conditional distributions for point processes (Q749041)

Let N be a point process on a Polish space E, and let \(C'\) be the compound Campbell measure of N. For A a subset of E, let \(N_ A()=N(\cdot \cap A)\) denote the restriction of N to A. Using a standard representation of certain conditional distributions, the author obtains several theorems concerning distributions of point processes. These include a consistency theorem for conditional distributions \(P\{\) \(N\in (\cdot) | N_{B\cup C}\}\), where B and C are bounded sets, an expression for conditional probabilities \(P\{N_ a\in (\cdot) | N_{B^ c}\}\) in terms of the Gibbs measure of N and some properties of thinned point processes and Cox processes. The results are very similar to those in \textit{O. Kallenberg}, Random measures. 3 rd ed. (1983; Zbl 0544.60053)].