Berry-Ess\'een bound for complex Wiener-It\^{o} integral

TL;DR

This paper derives Berry-Esséen bounds for complex Wiener-Itô integrals, simplifying conditions for the Fourth Moment Theorem and applying results to the complex Ornstein-Uhlenbeck process.

Contribution

It provides explicit Berry-Esséen bounds for complex Wiener-Itô integrals and simplifies the contraction condition in the complex Fourth Moment Theorem.

Findings

Established upper and lower bounds in Wasserstein distance

Simplified the contraction condition for the complex Fourth Moment Theorem

Derived an optimal Berry-Esséen bound for a complex Ornstein-Uhlenbeck process statistic

Abstract

For complex multiple Wiener-It\^{o} integral, we present Berry-Ess\'een upper and lower bounds in terms of moments and kernel contractions under the Wasserstein distance. As a corollary, we simplify the previously known contraction condition of the complex Fourth Moment Theorem. Additionally, as an application, we explore the optimal Berry-Ess\'een bound for a statistic associated with the complex-valued Ornstein-Uhlenbeck process.