## Electronic Journal of Probability

### Branching-stable point processes

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 119, 26 pp.

Dates
Accepted: 6 November 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067225

Digital Object Identifier
doi:10.1214/EJP.v20-4158

Mathematical Reviews number (MathSciNet)
MR3425539

Zentralblatt MATH identifier
1328.60046

Rights