Branching-stable point 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...

Statistical Inference for Stable Point Processes

Branching Processes

Galton-Watson processes were introduced by Francis Galton in 1889 as a simple mathematical model for the propagation of family names. They were reinvented by Leo Szilard in the late 1930s as models for the proliferation of free neutrons in a nuclear fission reaction. Generalizations of the extinction probability formulas that we shall derive below played a role in the calculation of the critica...