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Branching-stable point processes

Giacomo Zanella ; Sergei Zuyev (Institutionen för matematiska vetenskaper, matematisk statistik)
Electronic Journal of Probability (1083-6489). Vol. 20 (2015), p. artikel nr 119.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: stable distribution, discrete stability, Levy measure, point process, Poisson process, Cox process, distributions, stability, Mathematics

Denna post skapades 2015-11-27. Senast ändrad 2015-12-15.
CPL Pubid: 226472


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Institutionen för matematiska vetenskaper, matematisk statistik (2005-2016)



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