On the Gaussian approximation of vector-valued multiple integrals

Salim Noreddine and Ivan Nourdin

Journal of Multivariate Analysis, 2011, vol. 102, issue 6, 1008-1017

Abstract: 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 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 phenomenon. To reach this goal, we offer two results of a 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 -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.

Keywords: Central; limit; theorem; Cumulants; Malliavin; calculus; Multiple; integrals; Ornstein-Uhlenbeck; semigroup (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (3)

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