An improved complex fourth moment theorem

TL;DR

This paper improves the conditions for the complex Fourth Moment Theorem, providing sharper bounds and exploring asymptotic properties of related stochastic processes, with some bounds proven to be optimal.

Contribution

It introduces an improved contraction condition for the complex FMT and derives a new Berry-Esséen bound, including multivariate cases linked to index ordering.

Findings

Enhanced contraction condition for complex FMT

Optimal Berry-Esséen bounds in special univariate cases

Novel relation between multivariate bounds and index order

Abstract

For a series of univariate or multivariate complex multiple Wiener-It\^o integrals, we appreciably improve the previously known contractions condition of complex Fourth Moment Theorem (FMT) and present a fourth moment type Berry-Ess\'een bound under Wasserstein distance. Note that in some special cases of univariate complex multiple Wiener-It\^o integral, the Berry-Ess\'een bound we acquired is optimal. A remarkable fact is that the Berry-Ess\'een bound of multivariate complex multiple Wiener-It\^o integral is related to the partially order of the index of the complex multiple Wiener-It\^o integral, which has no real counterparts as far as we know. As an application, we explore the asymptotic property for the numerator of a ratio process which originates from the classical Chandler wobble model.