An Edgeworth expansion for functionals of Gaussian fields and its applications

This paper is concerned with the rate of convergence in the normal approximation of the sequence $\left\{F_n\right\}$, where each $F_n$ is a functional of an infinite-dimensional Gaussian field. We develop new and powerful techniques for computing the exact rate of convergence in distribution with respect to the Kolmogorov distance. As a tool for our works, the Edgeworth expansion of general orders, with an explicitly expressed remainder, will be obtained, and this remainder term will be controlled to find upper and lower bounds of the Kolmogorov distance in the case of an arbitrary sequence $\left\{F_n\right\}$. As applications, we provide the optimal fourth moment theorem of the sequence $\left\{F_n\right\}$ in the case when $\left\{F_n\right\}$ is a sequence of random variables living in a fixed Wiener chaos or a finite sum of Wiener chaoses. In the former case, our results show that the conditions given in this paper seem more natural and minimal than ones appeared in the previous works.