Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space

Stochastic evolutionary systems of additive functional type, described by processes with locally independent increments, are considered with Markov switching in an asymptotic split state space having a stoppage state. The average and diffusion approximation limit theorems are established in both single and double merging. The proofs of these results are obtained using a singular perturbation approach of linear reducible--invertible operators and the tightness of processes. Particular cases of these systems including integral functionals, dynamic systems, storage processes and compound Poisson processes are also considered. The application of limit theorems in reliability and reward problems is discussed.