Invariance principle for integral type functionals

Let {Xnj, n [greater-or-equal, slanted] 1, j[greater-or-equal, slanted]1} be a doubly indexed array of random variables, and let [tau]n= {[tau]n(t), 0<=t<=1},n[greater-or-equal, slanted]1, be a sequence of stochastic processes. We assume that the processes {[tau]n, n>=1} are nondecreasing, have left limits and are right continuous. Let Sni=[Sigma]ik=1Xnk, V2ni =[Sigma]ik=1X2nk,k[greater-or-equal, slanted]1,n[greater-or-equal, slanted]1. Suppose f,fn,n[greater-or-equal, slanted]1, are functions defined on [0,[infinity])x(-[infinity],[infinity]), and define , , 0<=t<=1, where {W(t), 0<=t<=1} is a standard Wiener process on the space D[0, 1]. The paper presents sufficient conditions which ensure the weak convergence, in the space D[0, 1] of {Zn(t), 0<=t<=1} to {Z(t), 0<=t<=1} as n-->[infinity].