On the Gaussian approximation of vector-valued multiple integrals

By combining the ndings 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 Fn 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 Fn to zero; (ii) the covariance matrix of Fn to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenom. To reach this goal, we o er two results of di erent nature. The rst one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F , when F is a Rd-valued random vector whose components are multiple integrals of possibly di erent 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.