The optimal third moment theorem

Abstract

This paper is concerned with the exact rate of convergence of the distribution of the sequence {Fn}, where each Fn is a functional of an infinite-dimensional Gaussian field. Nourdin and Peccati (2015) obtain a quantitative bound to complement the fourth moment theorem, by which a sequence in a fixed Wiener chaos converges in law to normal distribution if and only if the fourth cumulant converges to zero. Recently, Neufcourt and Viens (2016) show that a third moment theorem holds in the case of quadratic variations for stationary Gaussian sequences. In this paper, we find a sufficient condition on the optimal third moment theorem in the general case beyond the case of quadratic variations for stationary Gaussian sequences. As a main tool for our works, the recent results in Kim and Park (2018) will be used. As applications, we provide the optimal third moment theorem in the case when {Fn} is a sequence of sum of two integrals with respect to a fractional Gaussian noise.