Branching-stable point measures and processes

Branching-stable point processes

The notion of stability can be generalised to point processes by defining the scaling operation in a randomised way: scaling a configuration by t corresponds to letting such a configuration evolve according to a Markov branching particle system for -log t time. We prove that these are the only stochastic operations satisfying basic associativity and distributivity properties and we thus introdu...

Stable Marked Point Processes

In many contexts, such as queueing theory, spatial statistics, geostatistics and meteorology, data are observed at irregular spatial positions. One model of this situation is to consider the observation points as generated by a Poisson Process. Under this assumption, we study the limit behavior of the partial sums of the Marked Point Process {(ti, X(ti))}, where X(t) is a stationary random fiel...

A note on stable point processes occurring in branching Brownian motion

We call a point process Z on R exp-1-stable if for every α, β ∈ R with e + e = 1, Z is equal in law to TαZ + TβZ , where Z ′ is an independent copy of Z and Tx is the translation by x. Such processes appear in the study of the extremal particles of branching Brownian motion and branching random walk and several authors have proven in that setting the existence of a point process D on R such tha...

Branching Processes, and Random-cluster Measures on Trees

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of ‘boundary condition’, namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of randomcluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is enco...

Statistical Inference for Stable Point Processes