Thinning of point processes, revisited (Q1286666)

Let \(N,N_1,N_2\) be simple point processes on an LCCB space \((E,{\mathcal E})\) such that \(N=N_1+N_2\) and let \(p(\cdot)\) be a measurable function with \(0<p(\cdot)<1\) on \((E,{\mathcal E})\). The paper states that any two of the following statements yield the other two: (i) \(N\) is a Poisson process; (ii) \(N_1\) is the \(p(\cdot)\)-thinning of \(N\) and \(N_2\) is the \((1-p(\cdot))\)-thinning of \(N\); (iii) \(N_1\) and \(N_2\) are independent; (iv) \(N_1\) and \(N_2\) are Poisson processes with respect to the common filtration \(\{F(A),A \in{\mathcal E}\}\), where \(F(A):=\sigma \{N_1(B),N_2(B),\;B\in{\mathcal E}, B\subseteq A\}\). Only the implication (ii)+(iii)\(\Rightarrow\)(i) and (i)+(iv)\(\Rightarrow\)(iii) are new [for the other implications see for instance \textit{E. Çinlar}, in: Stochastic point processes, Statist. Analysis Theory Appl., 549-606 (1972; Zbl 0267.60056), \textit{K.-H. Fichtner}, Math. Nachr. 68, 93-104 (1975; Zbl 0328.60033) and \textit{A. F. Karr}, J. Multivariate Anal. 16, 368-392 (1985; Zbl 0584.62139)]. The first implication is shown by using Laplace functionals. The second implication uses the fact that for simple Poisson processes \(N_i\) the \(\pi\)-system \(\{\{N_i(A)=0\}\), \(A\in{\mathcal E}\) and \(A\) bounded\} generates the \(\sigma\)-field \(\sigma\{N_i(A),A\in{\mathcal E}\}\). By showing that the events \(\{N_1(A_1) =0\}\) and \(\{N_2(A_2)=0\}\) are independent for all bounded sets \(A_1,A_2\in {\mathcal E}\), this implies that the processes \(N_1\) and \(N_2\) are independent.