Limit theorems for a class of critical superprocesses with stable branching

Abstract

We consider a critical superprocess $\{X;{\mathbf{P}}_{\mu }\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index ${\gamma }_0>1$. We first show that, under some conditions, ${\mathbf{P}}_{\mu }(\Vert X_t\Vert \ne 0)$ converges to $0$ as $t\to \infty$ and is regularly varying with index $({\gamma }_0-1{)}^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t(f);{\mathbf{P}}_{\mu }(\cdot |\Vert X_t\Vert \ne 0)\}$, after appropriate rescaling, converges weakly to a positive random variable ${\mathbf{z}}^{({\gamma }_0-1)}$ with Laplace transform $E[e^{-u{\mathbf{z}}^{({\gamma }_0-1)}}]=1-(1+u^{-({\gamma }_0-1)}{)}^{-1/({\gamma }_0-1)}.$