On the convergence to the multiple Wiener-Ito integral

Abstract

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in ${\mathcal{C}}_0([0,T])$. Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function $f\in L^2([0,T{]}^n)$. We prove also the weak convergence in the space ${\mathcal{C}}_0([0,T])$ to the second order integral for two important families of processes that converge to a standard Brownian motion.