  Let $W=(W_t)_{t\ge0}$ be a supercritical $\alpha$-stable Dawson-Watanabe
process (with $\alpha\in(0,2]$) and $f$ be a test function in the domain of
$-(-\Delta)^{\frac \alpha2}$ satisfying some integrability condition. Assuming
the initial measure $W_0$ has a finite positive moment, we determine the
long-time asymptotic of all orders of $W_t(f)$. In particular, it is shown that
the local behavior of $W_t$ in long-time is completely determined by the
asymptotic of the total mass $W_t(1)$, a global characteristic.
