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

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