Gaussian Limits for Vector-valued Multiple Stochastic Integrals

Abstract

We establish necessary and sufficient conditions for a sequence of d-dimensional vectors of multiple stochastic integrals $\mathbf{F}_{d}^{k} = (F_{1}^{k}, \dots, F_{d}^{k})$, $k\geq 1$, to converge in distribution to a d-dimensional Gaussian vector $\mathbf{N}_{d} = (N_{1}, \dots, N_{d})$. In particular, we show that if the covariance structure of F _d ^k converges to that of N _d , then componentwise convergence implies joint convergence. These results extend to the multidimensional case the main theorem of [10].