The optimal third moment theorem

• Published in: Journal of the Korean Statistical Society Vol. 48; no. 4; pp. 636 - 658
• Main Authors: Kim, Yoon Tae, Park, Hyun Suk
• Format: Journal Article
• Language: English
• Published: Singapore Elsevier B.V 01.12.2019; Springer Singapore; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Berry–Esseen bound; Central limit theorem; Edgeworth expansions; Fourth moment theorem; Malliavin calculus; Multiple stochastic integral; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Stein’s method; Third moment theorem; 통계학; secondary; Edgeworth expansions; Stein’s method; Central limit theorem; Third moment theorem; Berry–Esseen bound; Multiple stochastic integral; Malliavin calculus; primary; Fourth moment theorem; Berry-Esseen bound; primary 60H07; secondary 60F25
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1016/j.jkss.2019.03.003

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.