Weak convergence of a difference additive functionals of Markov chains to a local times of random processes

We consider difference additive functionals defined on a family of Markov chains that weakly converge to a continuous time Markov process. We introduce an approach that allows one to give sufficient conditions for such a functionals to converge in distribution to a local times of the limiting process, formulated in the terms of their characteristics (i.e. expectations). This result generalizes E.B.Dynkin's theorem on convergence of W-functionals of a time homogeneous Markov process