We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

Approximate results are obtained for the probability distribution of the number of mutant substitutions over a given number of generations. The distribution is a compound Poisson distribution having a variance which is greater than the mean. The substitutions occur in clusters. The most extreme case of clustering is studied in this paper, which assumes that there is no recombination between mutant nucleotides.

This paper deals with the preservation of strong forms of mixing of point processes under the clustering operation. Strong, ϕ -, and *-mixing possess an increasing degree of uniformity of their asymptotic independence, and this turns out to be a significant factor in their preservation. In particular, it is indicated that ϕ-mixing may be maintained only under very stringent conditions (bounded clusters), whereas strong mixing is maintained under considerably milder conditions.

Several authors have noted that simple models for the evolution of a reproducing and spatially distributed population have no limiting distribution, although a Poisson process in statistical equilibrium has sometimes been implicitly assumed. It is shown that, even when a mechanism for restricting population density is postulated, a Poisson process is usually impossible to achieve, essentially because of an assumption of independent displacements. When this assumption is abandoned, a Poisson process is possible, at least for some highly idealised models.

Individuals situated on the infinite line breed and produce offspring that independently migrate a distance that is a random variable depending on the number in each family. It is shown that under very general conditions the individuals in the nth generation will tend to form clumps.

By using a cluster representation of the mutually-exciting processes their probability generating functional is derived. This is used to obtain some counting and interval properties of the process. The theoretical results are applied to a numerical example.

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.