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


discrete stability

random measure

Poisson process

point process

lévy measure

branching process




stable distribution


Giacomo Zanella

The University of Warwick

Sergey Zuev

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematical Statistics

Electronic Journal of Probability

10836489 (eISSN)

Vol. 20 119

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