DEFINITIONS AND RESULTSBy analogy with the theory of summation (A-scheme) of independent random variables (i.r.v.'s), one dimensional limit distributions of their products (A4-scheme) were investigated in the papers [1-5, 12, 13]. Having in mind the well-known invariance principles in the .A-scheme (see, for instance [6,8, 10,11]) in the present paper we start to deal with functional limit laws for step functions defined via products of i.r.v.'s. Of course, the general results and concepts on convergence of measures in functional spaces presented in the books [6,11 ] k<~yn(u) in the space D. The projection rrl: D ---> R defined by niX = X(1), X ~ D is continuous and therefore convergence in distribution of Yn (I) will be contained in the stated functional problem. Observe that instead of (1), the following more restrictive condition of A4-infinite~i,,,,iity maxl~i E) = o(I) has been used in ale papers [4,5,12,13], nevertheless, the results and the auxilliary calculations, which we will quote in this paper, remain valid under assumption (1).It appears that when dealing with convergence of the finite dimensional distributions of Yn(u) it is more convenient to use some modified convergence which is a little stronger than the weak convergence. This obstacle has appeared in the theorems concerning the random variable Yn(1) (see [1-5, 12, 13]). Moreover, avoiding influence of the signs of the first few multipliers Gk (this will be explained in the Concluding remark), instead of Yn(u), we will investigate convergence of the process Z.(u) := Y~(v~(u)),
