Branching-stable point processes
Journal article, 2015

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 introduce the notion of branching-stable point processes. For scaling operations corresponding to particles that branch but do not diffuse, we characterise stable distributions as thinning-stable point processes with multiplicities given by the quasi-stationary (or Yaglom) distribution of the branching process under consideration. Finally we extend branching-stability to continuous random variables with the help of continuous branching (CB) processes, and we show that, at least in some frameworks, branching-stable integer random variables are exactly Cox (doubly stochastic Poisson) random variables driven by corresponding CB-stable continuous random variables.

Cox process

Mathematics

discrete stability

random measure

Poisson process

point process

lévy measure

branching process

cb-process

distributions

stability

stable distribution

Author

Giacomo Zanella

The University of Warwick

Sergey Zuev

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematical Statistics

Electronic Journal of Probability

1083-6489 (ISSN)

Vol. 20 119

Subject Categories

Mathematics

DOI

10.1214/EJP.v20-4158

More information

Latest update

4/1/2021 1