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Branching-stable point processes

Journal article
Authors Giacomo Zanella
Sergei Zuyev
Published in Electronic Journal of Probability
Volume 20
Pages artikel nr 119
ISSN 1083-6489
Publication year 2015
Published at Department of Mathematical Sciences, Mathematical Statistics
Pages artikel nr 119
Language en
Links dx.doi.org/10.1214/EJP.v20-4158
Keywords stable distribution, discrete stability, Levy measure, point process, Poisson process, Cox process, distributions, stability, Mathematics
Subject categories Mathematics

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.

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