A point process approach to filtered processes (Q1762883)

Let us start with a definition of a filtered process. Let \(\lambda\) be a nonnegative \(\sigma\)-finite measure on the real axis \(\mathbb R\) and \(\mu\) be a probability measure on \(\mathbb R_+\). Let \(\mathbb P\) be a Poisson point process on \(\mathbb R\times\mathbb R\) of intensity measure \(\lambda \otimes \mu\), and \(h: \mathbb R^2\times \mathbb R_+ \mapsto \mathbb R_+\) be a map satisfying the condition: For any \((\tau, u)\in \mathbb R^2\) and any \(t\in \mathbb R_+\), \(t<\tau\) yields \(h(t,\tau, u)=0\). Then the following process \(M_t = \sum_{(\tau, u)\in \mathbb P} h(t, \tau, u)\) is called the (Poisson) filtered process of \(\mathbb P\) by the answer \(h\). An example of a filtered process is the classical counting Poisson process of constant intensity \(\lambda\) and intensity measure \(\pi = \lambda\, d\tau\otimes \delta_1\) on \(\mathbb R^2\), and the answer \(h(t, \tau, u) = u\, I_{[\tau, +\infty[}(t)\), where \(I_A(t)\) is the indicator function of \(A\). In Section 2 the author considers the quantity \(P(M_t\geq x)\), \(x>0\), and gives some calculations in a simplified model of river flow. A spatial version of filtered point processes is also studied, and using a Boolean model the author computes bounds for probability of dryness in compound rainfall processes [see also \textit{I. Molchanov}, ``Statistics of the Boolean models for practitioners and mathematicians'' (1997; Zbl 0878.62068)]. In Section 3 the author defines spatial compound filtered point processes. In Section 4 simulations of filtered point processes are considered.