Limit Theorems for Some Critical Superprocesses

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 $t\to \mathrm{\infty }$, of $t^{-1}⟨f,X_t⟩$ exists in distribution and we identify this limit. Finally, we also establish, under some conditions, a central limit theorem for $⟨f,X_t⟩$ conditioned on non-extinction at time $t$.