Branching-stable point processes

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.