Poisson approximation of processes with locally independent increments with Markov switching

In this paper, the weak convergence of additive functionals of processes with locally independent increments and with Markov switching in the scheme of the Poisson approximation is proved. For the relative compactness, a method proposed by R. Liptser for semimartingales is used with a modification, where we apply a solution of a singular perturbation problem instead of the ergodic theorem.