Normal approximation when a chaos grade is greater than two

Abstract

We derive an upper bound on a probabilistic distance for a normal approximation when the chaos grade of an eigenfunction of Markov diffusion generator L is greater than 2. When a chaos grade is strictly greater than 2, the upper bound, given by Bourguin et al. (2019), does not guarantee that Fn converges in distribution to a standard Gaussian distribution even when the fourth cumulant of Fn converges to 0. This means that the fourth moment theorem, discovered by Nualart and Peccati (2005), does not work. In this paper, we develop a new technique to obtain an upper bound for which the fourth moment theorem works when a chaos grade is strictly greater than 2.