We study higher-order moment measures of heavy-tailed renewal models, including a renewalpoint process with heavy-tailed inter-renewal distribution and its continuous analog, theoccupation measure of a heavy-tailed Lévy subordinator. Our results reveal that theasymptotic structure of such moment measures are given by explicit power-law densityfunctions. The same power-law densities appear naturally as cumulant measures of certainPoisson and Gaussian stochastic integrals. This correspondence provides new and extendedresults regarding the asymptotic fluctuations of heavy-tailed sources under aggregation,and clarifies existing links between renewal models and fractional random processes.

We consider a dynamic multilevel population or information system. At each level individuals or information units undergo a Galton–Watson-type branching process in which they can be replicated or removed. In addition, a collection of individuals or information units at a given level constitutes an information unit at the next higher level. Each collection of units also undergoes a Galton–Watson branching process, either dying or replicating. In this paper, we represent this multilevel branching model as a measure-valued stochastic process, study its moment structure, identify the limiting continuous-state approximation and analyse the long-time behavior in both non-critical and critical cases. For example, we obtain an asymptotic expression for the extinction probability for the total population mass process and an analogue of Yaglom's conditioned limit theorem in the critical case.

We consider a multitype branching random walk with independent Poisson random fields of each type of particle initially. The existence of limiting random fields as the generation number, is studied, when the intensity of the initial field is renormalized in such a way that the mean measures converge. Spatial laws of large numbers and central limit theorems are given for the limiting random field, when it is non-trivial.

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.