  In this paper, we establish a central limit theorem for a large class of
general supercritical superprocesses with immigration with spatially dependent
branching mechanisms satisfying a second moment condition. This central limit
theorem extends and generalizes the results obtained by Ren, Song and
Zhang[2015]. We first give law of large numbers for supercritical
superprocesses with immigration since there is few convergence result on
immigration superprocesses, then based on these results, we establish the
central limit theorem.
