  In this paper, we study the asymptotic behavior of a supercritical
$(\xi,\psi)$-superprocess $(X_t)_{t\geq 0}$ whose underlying spatial motion
$\xi$ is an Ornstein-Uhlenbeck process on $\mathbb R^d$ with generator $L =
\frac{1}{2}\sigma^2\Delta - b x \cdot \nabla$ where $\sigma, b >0$; and whose
branching mechanism $\psi$ satisfies Grey's condition and some perturbation
condition which guarantees that, when $z\to 0$, $\psi(z)=-\alpha z + \eta
z^{1+\beta} (1+o(1))$ with $\alpha > 0$, $\eta>0$ and $\beta\in (0, 1)$. Some
law of large numbers and $(1+\beta)$-stable central limit theorems are
established for $(X_t(f) )_{t\geq 0}$, where the function $f$ is assumed to be
of polynomial growth. A phase transition arises for the central limit theorems
in the sense that the forms of the central limit theorem are different in three
different regimes corresponding the branching rate being relatively small,
large or critical at a balanced value.
