  We study the averaging principle for a family of multiscale stochastic
dynamical systems. The fast and slow components of the systems are driven by
two independent stable L\'evy noises, whose stable indexes may be different.
The homogenizing index $r_0$ of slow components has a relation with the stable
index $\alpha_1$ of the noise of fast components given by
$0<r_0<2-2/{\alpha_1}$. By first studying a nonlocal Poisson equation and then
constructing suitable correctors, we obtain that the slow components weakly
converge to a L\'evy process as the scale parameter goes to zero.
