  We study the limit behavior of weighted Bergman kernels on a sequence of
domains in a complex space $\mathbb{C}^N$, and show that under some conditions
on domains and weights, weighed Bergman kernels converge uniformly on compact
sets. Then we give a weighted generalization of the theorem given in [S, p.
38], highlighting some special property of the domains, on which the weighted
Bergman kernels converge uniformly. Moreover we will show that convergence of
weighted Bergman kernels implies this property, which will give a
characterization of the domains, for which the inverse of Ramadanov theorem
holds.
