Weak convergence to a class of multiple stochastic integrals

Abstract

ABSTRACTLet Q ∈ C((0, 1]1 + m) and let {ξn, j: 1 ⩽ j ⩽ n, n ⩾ 1} be a triangular array of independent zero-mean random variables. In this paper, we show that for every integer m ⩾ 1, the processes Xmn(Q) = {Xnm(t, Q), 0 ⩽ t ⩽ 1}, n ⩾ 1 defined byconverges weakly in C[0, 1], as n → ∞, to a multiple Wiener integral under some suitable conditions on the function Q and array {ξn, j}. This convergence includes the approximating characterization of many popular self-similar processes such as fractional Brownian motion, sub-fractional Brownian motion, Rosenblatt process, and more general Hermite processes.