发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Big Data, Baoshan University; LMAM, School of Mathematical Sciences, Peking University（中国科学院数学与系统科学研究院; 保山学院大数据学院; 北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究复多重Wiener-Itô积分的密度正则性与优化高斯逼近，提出核判据和显式密度公式，证明内在非高斯项与协方差失配之和在多种距离下达到最优速率，并应用于复分数Ornstein-Uhlenbeck过程的最小二乘估计。
AI 中文摘要
我们研究复多重Wiener-Itô积分的密度正则性与优化高斯逼近。首先给出绝对连续性的核判据，并推导密度及其导数的显式公式，以及有限阶Sobolev估计。我们的主要定量结果表明，内在非高斯项与协方差失配之和在每一个固定Sobolev范数以及全变差、Kolmogorov和$1$-Wasserstein距离下同时给出最优速率。内在项由两个复三阶矩和复四阶矩缺陷决定，并等价地由复核收缩描述。在这些渐近结果中不施加额外的Malliavin非退化假设。作为应用，我们获得复分数Ornstein--Uhlenbeck过程最小二乘估计中归一化分子的显式最优速率。
英文摘要
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.
