CENTRAL LIMIT THEOREM FOR WEIGHTED MARTINGALES WITH APPLICATIONS R. Norvaisa UDC 519.21 In this note the following problem is considered. Suppose given a sequence of martin- gales (Mn) and a sequence of predictable processes (wn) with nonincreasing trajectories. Under what conditions does one have weak convergence of the sequence of processes (Xn) having the form X. (t) = w. (t) Mn (t), te[O, T], ( 1 ) in the Banach space D = D([0, T]) with the uniform norm? As a rule the trajectories of the processes w n do not belong to the space D. But the conditions given in the node (cf. Theo- rems 1 and 3) do not exclude the case w n = i and thus let one get convergence simply for a sequence of martingales. The assertions obtained are later applied to questions of the asymptotic behavior of processes connected with empirical ones. i. Introduction It is known that a stochastic process similar to an empirical one considered as a map with values in a space of functions B, B~R [~ is nonseparable-valued and nonmeasurable (with respect to the Borel o-algebra) in the topology of uniform convergence~ Hence here it is necessary to apply a more general construction of weak convergence. Thus, for example, Anderson and Dobric [6] proved necessary and sufficient conditions for the validity of the CLT in the space of bounded functions, generalizing those obtained earlier in Dudley and Philipp [8] for empirical processes. We note that these conditions are a generalization of the criterion (3) used below. To this end they applied the theory of weak convergence of non- separable-valued and nonmeasurable functions which at the present time has received its greatest development in Hoffmann-J~rgensen [15]. In the present note we adhere to the direc- tion in which the problem of measurability is eliminated by the choice of a special o-algebra (cf., e.g~, [ii, 12]). Moreover, we assume that the processes have special structure, the most important part of which is the martingale property. It is necessary to note that at the present time there is a vast collection of papers devoted to the CLT (still called the func- tional CLT, invariance principle), in whose proof the martingale properties are used. It is well-known that the numerous applications are the reason for this. Here we only recall the papers of Rebolledo [24] and Helland [14], Liptser and Shiryaev [18], and Jacod [17] to whose theme the content of the present note is closest. One can find additional information on the CLT for martingales in [13, 27]. The situation whose generalization is the basic result often occurs in studies devoted to the asymptotic behavior of empirical processes and those connected with them. We have in mind the problem of the asymptotic behavior of weighted statistics. Some necessary points in the study of weighted statistics are discussed, for example, in [7] which is a paper we know. The basic result is applied as an illustration to empirical processes and also to the martingale part of the Doob-Meyer decomposition of this process. In sum the assertions ob- tained generalize (or improve) in certain respects the results of Shorack [21], Gaenssler [ii] (cf. also [12]), Aki [i, 2], Khmaladze [28]. With respect to applications the ideology of the note is close to the results of Ai-Hussaini and Elliot [4, 5], Jacobsen [16], Grigel- ionis and Mikulyavichyus [26]. But by virtue of the presence of the weight function w n in (i) the process X n itself generally stops being a martingale. In this, in particular, the note differs from the papers mentioned. 2. Notation and Terminology Let D = D([0, T]), T < ~ be the space of functions on the interval [0, T] which are right continuous and have left limits with values in R. We shall consider it as a (nonseparable) Vilnius University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Mate- matikos Rinkinys), Vol. 29, No. 4, pp. 754-772, October-December, 1989. Original article submitted August 25, 1988. 370 0363-1672/89/2904-0370512.50 9 1990 Plenum Publishing Corporation