Weak Convergence for Semimartingales

This chapter discusses the conditions of weak convergence for semimartingales and some of their applications. It gives the general sufficient conditions of weak convergence to quasi-left-continuous semimartingales. In particular, it includes the weak convergence for processes with independent increments. The results will be applied to the case, where the hmit process is a generalized diffusion process. The classical results of weak convergence for empirical processes are also given. For the sake of simplicity, the chapter considers the real-valued processes only, most of the results still hold for R^d-valued processes. It presents theorems and lemmas for quasi-left-continuous semimartingale. The apply the general results for convergence in law of semimartingales is applied to a special case, where the limit process is a continuous Levy process. The chapter also gives sufficient conditions for convergence in law of locally square integrable martingales and semimartingales to a continuous Levy process.