Branching-stable point processes

TL;DR

This paper introduces the concept of branching-stable point processes, characterizes their properties, and connects them to Cox processes driven by continuous branching processes, expanding the understanding of stability in stochastic point configurations.

Contribution

It defines and characterizes branching-stable point processes, linking them to thinning-stable distributions and Cox processes driven by continuous branching processes.

Findings

Branching-stable point processes are uniquely characterized by their stability under Markov branching evolution.

Stable distributions with local branching are identified as thinning-stable with Yaglom distributions.

In some frameworks, $\\mathcal{F}$-stable integer variables are Cox processes driven by CB-stable variables.

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. 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.