Limit Theorems for Some Critical Superprocesses

Abstract: In this paper we establish some conditional limit theorems for some critical superprocesses $X={X_t, t\ge 0}$. First we identify the rate of non-extinction. Then we show that, for a large class of functions $f$, conditioned on non-extinction at time $t$, the limit, as $t\to\infty$, of $t^{-1}\langle f, X_t\rangle$ exists in distribution and we identify this limit. Finally, we also establish, under some conditions, a central limit theorem for $\langle f, X_t\rangle$ conditioned on non-extinction at time $t$.