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Register a new userLimit theorems for a class of critical superprocesses with stable branching
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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 $E[e^{-umathbf z^{(gamma_0-1)}}]=1-(1+u^{-(gamma_0-1)})^{-1/(gamma_0-1)}.$
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In this paper, we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yagloms exponential limit law for critical superprocesses.
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