Invariance principle for integral type functionals

AbstractLet {Xnj, n ⩾ 1, j⩾1} be a doubly indexed array of random variables, and let τn= {τn(t), 0≤t≤1},n⩾1, be a sequence of stochastic processes. We assume that the processes {τn, n≥1} are nondecreasing, have left limits and are right continuous. Let Sni=Σik=1Xnk, V2ni =Σik=1X2nk,k⩾1,n⩾1. Suppose f,fn,n⩾1, are functions defined on [0,∞)×(−∞,∞), and define Zn(t)=∑i⩾τn(t)fn(V2n,i−1, Sn,i−1)Xni, Z(t)=∫t0f(s, W(s)) d W (s), 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→∞