Open Access
2015 Branching-stable point processes
Giacomo Zanella, Sergei Zuyev
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Electron. J. Probab. 20: 1-26 (2015). DOI: 10.1214/EJP.v20-4158


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.


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Giacomo Zanella. Sergei Zuyev. "Branching-stable point processes." Electron. J. Probab. 20 1 - 26, 2015.


Received: 4 March 2015; Accepted: 6 November 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1328.60046
MathSciNet: MR3425539
Digital Object Identifier: 10.1214/EJP.v20-4158

Primary: 60E07
Secondary: 60G55 , 60J68 , 60J85

Keywords: branching process , CB-process , Cox process , discrete stability , Lévy measure , point process , Poisson process , random measure , stable distribution

Vol.20 • 2015
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