Convergence to a Semimartingale

Here comes the third—and last—step in our exposition of limit theorems. Not only are the pre-limiting processes Xn arbitrary semimartingales, but the limit process X also is a semimartingale; not quite an arbitrary one, though: since the method is based here on convergence of martingales and on the relations between X and its characteristics, we need these characteristics to indeed characterize the distribution. ℒ(X) of X So, in most of the chapter, we will assume that ℒ(X) is the unique solution to the martingale problem associated with the characteristics of X, as introduced in Chapter III.