Weak convergence to a class of multiple stochastic integrals

Let 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 by Xm(n)(t,Q)=∑1≤j1,…,jm≤[nt]j1≠⋯≠jmnm∫j1-1nj1n⋯∫jm-1njmnQ[nt]n,r1,…,rmdr1⋯drm×∏k=1mξn,jk \begin{eqnarray*} \!\!\!\!X^{(n)}_m(t,Q)&=& \sum _{\substack{1\le j_1,\ldots,j_m\le [nt]\\ j_1\ne \cdots \ne j_m}}n^m\left(\int _{\frac{j_1-1}{n}}^{\frac{j_1}{n}}\cdots \int _{\frac{j_m-1}{n}}^{\frac{j_m}{n}} Q\left(\frac{[nt]}{n},r_{1},\ldots,r_m\right)dr_1\cdots dr_m\right)\nonumber\\ \!\!\!\!&&\times\prod \limits_{k=1}^m\xi _{n,j_k} \end{eqnarray*}converges 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.