Weighted approximations of partial sum processes in D[0, ∞). I

Let X1, X2, . . . be independent, identically distributed random variables with S(k) = X1+. . .+ Xk. We prove best possible weighted approximations of n-1/2S(nt), 0 t ≦1, by a standard Wiener process {W(i), 0 ≦ t < ∞} assuming the existence of two moments only for X1 and, consequently, obtain the weighted version of Donsker's theorem in D[0, 1] for the optimal class of weight functions. Considering functions h on [1, ∞) such that lim sup |W(t)|/h(t) < ∞ a.s. enables us to prove weighted approximations, and hence t→∞ also weak convergence, of weighted partial sum processes in D[1, ∞). In this case the admissible class of weight functions will be seen to be bigger than that for asymptotics on [0,1]. We show also that the class of weight functions for weighted sup- and Lp-functionals to converge in distribution is larger than that for weak convergence. Our proofs are based on Theorem of Major [17].