  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.
