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