Necessary and sufficient conditions for convergence of semimartingales and point processes. I (Q1058224)

In the paper the so-called extended weak convergence [cf. \textit{I. S. Helland}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 52, 251-265 (1980; Zbl 0411.60037) and the first author, Litov. Mat. Sb. 24, No.2, 97-112 (1984; see the preceding review, Zbl 0564.60004)] of a sequence of semimartingales is considered. Necessary and sufficient conditions for this type of convergence expressed in terms of predictable characteristics are obtained when the limit process is rather general, namely, a semimartingale. Similar problems for point processes are also discussed. The proofs of all these results are published in the second part of this paper.