On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals $F_n$ towards a centered Gaussian random vector $N$, with given covariance matrix $C$, is reduced to just the convergence of: $(i)$ the fourth cumulant of each component of $F_n$ to zero; $(ii)$ the covariance matrix of $F_n$ to $C$. The aim of this paper is to understand more deeply this somewhat surprising phenomenom. To reach this goal, we offer two results of different nature. The first one is an explicit bound for $d(F,N)$ in terms of the fourth cumulants of the components of $F$, when $F$ is a $\R^d$-valued random vector whose components are multiple integrals of possibly different orders, $N$ is the Gaussian counterpart of $F$ (that is, a Gaussian centered vector sharing the same covariance with $F$) and $d$ stands for the Wasserstein distance. The second one is a new expression for the cumulants of $F$ as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.