Weak Convergence in Poisson and Lévy Approximation Schemes

This chapter proposes a method for proving the weak convergence of random evolutionary systems, such as processes with locally independent increments and impulsive recurrent processes, that is significantly different from those used by other authors: the main task is to prove convergence of the predictable characteristics of the semimartingale, which are integral functionals that depend on the switching process. But the problem is that these predictable characteristics also depend on the previous values of the process itself. So, to prove weak convergence of the process, we have to prove weak convergence of its predictable characteristics that are dependent on the process. The proof of weak convergence of predictable characteristics is carried out similarly to the case of semi-Markov switching in the Poisson approximation scheme. The chapter considers the normalized random evolutionary systems under Levy approximation conditions.