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Limit theorems for a class of critical superprocesses with stable branching (1807.02837v2)
Published 8 Jul 2018 in math.PR
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, $\mathbf P_{\mu}(|X_t|\neq 0)$ 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 ${X_t(f);\mathbf P_\mu(\cdot||X_t|\neq 0)}$, after appropriate rescaling, converges weakly to a positive random variable $\mathbf z{(\gamma_0-1)}$ with Laplace transform $E[e{-u\mathbf z{(\gamma_0-1)}}]=1-(1+u{-(\gamma_0-1)}){-1/(\gamma_0-1)}.$