Berry-Esséen bound for complex Wiener-Itô integral

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.