Briefly, a Cox process is a natural extension of a Poisson process, considering the intensity measure of the Poisson process as a realization of a random measure. When a Cox process \(X\) is defined in the \(d\)-dimensional Euclidean space \(R^d\), it is usually specified by a random field \(Z(\xi)\geq 0\), \(\xi\in R^d\), so that the conditional distribution of \(X\) given \(Z\) is a Poisson point process on \(R^d\) with intensity function \(Z\). Many recent papers deal with simulation-based inference for new flexible model classes of such Cox processes [see the author and R. P. Waagepetersen, “Statistical inference and simulation for spatial point processes” (2004; Zbl 1044.62101)]).
This paper is concerned with shot noise Cox processes (SNCPs), i.e. when \(Z\) is of the form \(Z(\xi) = \sum_j \gamma_j k(c_j, \xi)\), where \(k(c_j,\cdot)\) is a kernel (i.e. a density function for a continuous \(d\)-dimensional random variable) and the \((c_j, \gamma_j)\in R^d\times (0, \infty)\) are the points of a Poisson point process \(\Phi\) on \(R^d\times (0, \infty)\). This is a rich class of Cox process models which includes Neyman-Scott processes and Poisson-gamma processes as special cases. The focus in the paper is on the probabilistic aspects of SNCPs with a view to statistical applications.
The paper is organized as follows. Section 2 provides some background material. Section 3 concerns results for the summary statistics and the reduced Palm distributions of SNCPs. Section 4 studies different simulation algorithms for simulation of an SNCP within a bounded window \(W\), and for conditional simulation of \(\Phi\) given the restriction of an SNCP within \(W\). Finally, Section 5 deals with local and spatial Markov properties of SNCPs when the kernel has a bounded support.