Convergence of weighted partial sums when the limiting distribution is notnecessarily Radon

Let B-w be a non-separable Banach space of real-valued functions endowed wi th a weighted sup-norm. We consider partial sum processes as random functio ns with values in B-w. We establish weak convergence statements for these p rocesses via their weighted approximation in probability by an appropriate sequence of Gaussian random functions. The main result deals with convergen ce of distributions of certain functionals in the case when the Wiener meas ure is not necessarily a Radon measure on B-w. (C) 1999 Elsevier Science B. V. All rights reserved.