Optimal Gaussian approximation for complex Wiener-Itô integrals
Abstract
We study density regularity and optimal Gaussian approximation for complex multiple Wiener-Itô integrals. We first give a kernel criterion for absolute continuity and derive explicit formulas for the density and its derivatives, together with finite-order Sobolev estimates. Our main quantitative result shows that the sum of an intrinsic non-Gaussian term and the covariance mismatch gives the optimal rate simultaneously in every fixed Sobolev norm and in total variation, Kolmogorov and $1$-Wasserstein distances. The intrinsic term is determined by two complex third-order moments and the complex fourth-moment defect, and is equivalently described by complex kernel contractions. No additional Malliavin non-degeneracy assumption is imposed in these asymptotic results. As an application, we obtain explicit optimal rates for the normalized numerator in the least-squares estimator of a complex fractional Ornstein--Uhlenbeck process.
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BibTeX
@misc{chen2026optimal,
title = {{Optimal Gaussian approximation for complex Wiener-Itô integrals}},
author = {Huiping Chen and Yong Chen and Yong Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04532},
eprint = {2610.04532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04532},
}
RIS
TY - UNPB
TI - Optimal Gaussian approximation for complex Wiener-Itô integrals
AU - Huiping Chen
AU - Yong Chen
AU - Yong Liu
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04532
UR - https://arxiv.org/abs/2610.04532
AB - We study density regularity and optimal Gaussian approximation for complex multiple Wiener-Itô integrals. We first give a kernel criterion for absolute continuity and derive explicit formulas for the density and its derivatives, together with finite-order Sobolev estimates. Our main quantitative result shows that the sum of an intrinsic non-Gaussian term and the covariance mismatch gives the optimal rate simultaneously in every fixed Sobolev norm and in total variation, Kolmogorov and $1$-Wasserstein distances. The intrinsic term is determined by two complex third-order moments and the complex fourth-moment defect, and is equivalently described by complex kernel contractions. No additional Malliavin non-degeneracy assumption is imposed in these asymptotic results. As an application, we obtain explicit optimal rates for the normalized numerator in the least-squares estimator of a complex fractional Ornstein--Uhlenbeck process.
ER -