Critical cluster cascades

Abstract We consider a sequence of Poisson cluster point processes on $\mathbb{R}^d$ : at step $n\in\mathbb{N}_0$ the construction, centers have intensity $c/(n+1)$ for some $c>0$ , and each consists particles branching random walk up to generation n —generated by process with mean 1. show that this ‘critical cascade’ converges weakly, either limit equals void (extinction), or it has same c as critical cascade (persistence). obtain persistence if only Palm version outgrown is locally almost surely finite. This result allows us give numerous examples persistent cascades.