Branching-stable point processes
| Autor: | Zanella, Giacomo, Zuyev, Sergei |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. We characterise stable distributions with respect to local branching 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 random variables with the help of continuous branching (CB) processes, and we show that, at least in some frameworks, $\mathcal{F}$-stable integer random variables are exactly Cox (doubly stochastic Poisson) random variables driven by corresponding CB-stable continuous random variables. Comment: 31 pages. To appear in Electronic Journal of Probability |
| Databáze: | arXiv |
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