2015/03/04 by Giacomo Zanella, Zanella, Giacomo, Sergei Zuyev +1
Mathematics · #60J68 #60J85 #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Primary 60E07 #Probability (math.PR) #Secondary 60G55 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1503.01329
openalex publication_date 2015/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, F-stable integer random variables are exactly Cox (doubly stochastic Poisson) random variables driven by corresponding CB-stable continuous random variables.