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We develop a technique for obtaining the fourth moment bound on the normal approximation of F, where F is an Rd-valued random vector whose components are functionals of Gaussian fields. This study transcends the case of vectors of multiple stochastic integrals, which has been the subject of research so far. We perform this task by investigating the relationship between the expectations of two operators Γ and Γ*. Here, the operator Γ was introduced in Noreddine and Nourdin (2011) [On the Gaussian approximation of vector-valued multiple integrals. J. Multi. Anal.], and Γ* is a muilti-dimensional version of the operator used in Kim and Park (2018) [An Edgeworth expansion for functionals of Gaussian fields and its applications, stoch. proc. their Appl.]. In the specific case where F is a random variable belonging to the vector-valued multiple integrals, the conditions in the general case of F for the fourth moment bound are naturally satisfied and our method yields a better estimate than that obtained by the previous methods. In the case of d=1, the method developed here shows that, even in the case of general functionals of Gaussian fields, the fourth moment theorem holds without conditions for the multi-dimensional case.

Fourth Cumulant Bound of Multivariate Normal Approximation on General Functionals of Gaussian Fields