Poisson cluster processes are one of the most important classes of point process models (see Daley and Vere-Jones[7] and Møller; Waagepetersen[24]). They are natural models for the location of objects in the space, and are widely used in point process studies whether theoretical or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or Hawkes processes ...

The definition and the properties of a Gaussian point distribution, in contrast to the well-known properties of a Gaussian random field are discussed. Constraints for the number density and the two-point correlation function arise. A simple method for the simulation of this so-called Gauss-Poisson point process is given and illustrated with an example. A comparison of the distribution of galaxi...

The asymptotic theory for the sample mean of a marked point process in d dimensions is established, allowing for the possibility that the underlying Poisson point process is inhomogeneous. A novel local block bootstrap method for resampling inhomogeneous Poisson marked point processes is introduced, and its consistency is proven for the sample mean and related statistics. Finite-sample simulati...

the problems of sequential change-point have several important applications in quality control, signal processing, and failure detection in industry and finance. we discuss a bayesian approach in the context of statistical process control: at an unknown time $tau$, the process behavior changes and the distribution of the data changes from p0 to p1. two cases are considered: (i) p0 and p1 are fu...

We propose a weighted analogue of Ripley’s K-function for assessing the fit of point process models. The advantage of the proposed measure is that it can be used in situations where the null hypothesis is not a stationary Poisson model. We present its distributional properties for a spatial, two-dimensional inhomogeneous Poisson process and use it to assess the goodness-of-fit of two alternativ...

In 1971, Meyer showed how one could use the compensator to rescale a multivariate point process, forming independent Poisson processes with intensity one. Meyer’s result has been generalized to multi-dimensional point processes. Here, we explore generalization of Meyer’s theorem to the case of marked point processes, where the mark space may be quite general. Assuming simplicity and the existen...

For an arbitrary point of a homogeneous Poisson point process in a d-dimen-siona! Euclidean space, consider the number of Poisson points that have that given point as their r-th nearest neighbor (r = 1,2,...). It is shown that as d tends to infinity, these nearest neighbor counts (r = 1,2,...) are tid asymptotically Poisson with mean 1. The proof relies on Renyi's characterization of Poisson pr...

Consider the region L := {(x, y) : 0 ≤ y ≤ C log(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the percolation properties we place a Poisson point process of intensity λ on the region L. At each point of the process we centre a box of a random side length ρ. In case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of perco...