Convergence of Processes with Independent Increments

With this chapter, at last, we enter the subject which has given its name to the whole book. Our final aim is to prove convergence theorems for a sequence of semimartingales toward a semimartingale. We present the material through three successive steps, corresponding to Chapters VII, VIII and IX: firstly, the pre-limiting processes, as well of course as the limiting process, have independent increments; secondly, only the limiting process has independent increments; thirdly, the limiting process itself belongs to some rather broad class of semimartingales. This method of exposition, going from the particular to the general, surely leaves way to many redundancies; it also have some advantages: 1)It allows to put together in a single chapter most of the “old” results that are due to Lévy, Khintchine, Kolmogorov, Gnedenko, etc., although this chapter also contains some recent or new results.2)The independent increments case shows a very simple structure, from the probabilistic point of view.3)At the same time, most of the analytical difficulties are already present in this case. Henceforth, our presentation allows to single out the two kinds of problems, analytical and probabilistic.4)Finally, the simple structure of PII’s allows for various necessary and sufficient conditions for convergence, a fact that is untrue in general.