Electronic Journal of Probability

Branching-stable point processes

Giacomo Zanella and Sergei Zuyev

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 119, 26 pp.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G55: Point processes 60J85: Applications of branching processes [See also 92Dxx] 60J68: Superprocesses

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

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Zanella, Giacomo; Zuyev, Sergei. Branching-stable point processes. Electron. J. Probab. 20 (2015), paper no. 119, 26 pp. doi:10.1214/EJP.v20-4158. https://projecteuclid.org/euclid.ejp/1465067225

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