Limit theorems for a class of critical superprocesses with stable branching

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, $mathbf P_{mu}(|X_t| eq 0)$ converges to $0$ as $tto infty$ 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 ${X_t(f);mathbf P_mu(cdot||X_t| eq 0)}$, after appropriate rescaling, converges weakly to a positive random variable $mathbf z^{(gamma_0-1)}$ with Laplace transform $E[e^{-umathbf z^{(gamma_0-1)}}]=1-(1+u^{-(gamma_0-1)})^{-1/(gamma_0-1)}.$