  Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by
a Gaussian noise $W=\{W(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and
colored in space with correlation kernel $g(x,y)$. When $d\geq 3$, under the
condition that the correlation function $g(x,y)$ is bounded above by some
appropriate function $\bar{g}(x-y)$, we present the quenched and annealed
Strong Law of Large Numbers and the Central Limit Theorems regarding the
weighted occupation measure $\int_0^t X_s ds$ as $t\to \infty$.
