Central limit theorems for local martingales

Abstract

This paper is involved with the following problem. Given a sequence of local martingales, say (Mn), under which conditions on the quadratic variations ([M,]), can we state the convergence in distribution of the (M,) sequence towards a continuous gaussian martingale limit? "Convergence in distribution" means here the weak convergence of the (s sequence on the space D of right continuous and left hand limited functions, f ( M , ) being the probability measure induced on D by M n (i.e. the distribution or law of the M, process). In preceding works, the author has investigated an analogous problem for locally square integrable local martingales (in short, "locally square integrable martingales"). In such case we were interested in finding out conditions on the ( (M,)) sequence of associated increasing processes to insure the (M.)'s convergence in distribution. It is a well known fact (c.f. [-9]) that for a local martingale M the associated increasing process ( M ) exists if and only if M is locally square integrable. On the contrary, [M] always exists and, furthermore, [M] is easier to calculate than ( M ) when both processes exist. Thus the problem with which we will deal below is a very natural one. In the first paragraph, we will explain some notations. Paragraph two is devoted to state the main results of this paper. Proofs of these results are given in paragraph three. Paragraph four contains some particular cases of the main theorems. The last paragraph gives a complementary result for locally square integrabte martingales. The Appendix contains the recall of a classical Tightness Criterion used in the paper.