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Branching-stable point processes
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Zanella, Giacomo and Zuyev, Sergei (2015) Branching-stable point processes. Electronic Journal of Probability, 20 . 119. doi:10.1214/EJP.v20-4158
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Official URL: http://dx.doi.org/10.1214/EJP.v20-4158
Abstract
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.
| Item Type: | Journal Article | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science > Statistics | ||||
| Library of Congress Subject Headings (LCSH): | Branching processes, Semigroups, Harmonic analysis, Harmonic analysis, Point processes, Distribution (Probability theory) | ||||
| Journal or Publication Title: | Electronic Journal of Probability | ||||
| Publisher: | University of Washington. Dept. of Mathematics | ||||
| ISSN: | 1083-6489 | ||||
| Official Date: | 7 November 2015 | ||||
| Dates: |
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| Volume: | 20 | ||||
| Article Number: | 119 | ||||
| DOI: | 10.1214/EJP.v20-4158 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Open Access |
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