Stable central limit theorems for super Ornstein-Uhlenbeck processes

In this paper, we study the asymptotic behavior of a supercritical (xi,psi)-superprocess (X-t)(t >= 0) whose underlying spatial motion xi is an Ornstein-Uhlenbeck process on R-d with generator L = 1/2 sigma(2)Delta - bx . del where sigma, b > 0; and whose branching mechanism psi satisfies Grey's condition and a perturbation condition which guarantees that, when z -> 0, psi(z) = -alpha z -eta z(1+beta) (1 + o(1)) with alpha > 0, eta > 0 and beta is an element of (0,1). Some law of large numbers and (1+ beta)-stable central limit theorems are established for (X-t(f))(t >= 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 to the branching rate being relatively small, large or critical at a balanced value.