Limit theorems for some critical superprocesses

Abstract

Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathbb{P}}_{\mu}(\Vert X_{t}\Vert \ne0)=\nu^{-1}\langle\phi_{0},\mu\rangle$, where $\phi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator $L$ of the mean semigroup of $X$, and $\nu$ is a positive constant. Then we show that, for a large class of functions $f$, conditioning on $\Vert X_{t}\Vert \ne0$, $t^{-1}\langle f,X_{t}\rangle$ converges in distribution to $\langle f,\psi_{0}\rangle_{m}W$, where $W$ is an exponential random variable, and $\psi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the dual of $L$. Finally, if $\langle f,\psi_{0}\rangle_{m}=0$, we prove that, conditioning on $\Vert X_{t}\Vert \ne0$, $(t^{-1}\langle\phi_{0},X_{t}\rangle,t^{-1/2}\langle f,X_{t}\rangle )$ converges in distribution to $(W,G(f)\sqrt{W})$, where $G(f)\sim\mathcal{N}(0,\sigma_{f}^{2})$ is a normal random variable, and $W$ and $G(f)$ are independent.