AI 中文总结
该研究刻画了可变高斯希尔伯特空间下维纳混沌弱闭包，提出希尔伯特-斯坦提取方法，构造本原-福克分类，得到相关极限性质与矩定理，消除了幽灵阻碍以恢复极大高斯因子。
AI 中文摘要
当 underlying 高斯希尔伯特空间可变时，我们刻画固定维纳混沌中一致 L²有界向量的弱闭包。在一个已表示的子序列上，每个物理权重张量空间分解为可分解闭张成及其本原正交补。本原块评估可扩展为酉加权福克表示，因此第 q 阶混沌的弱极限恰好是由 q 的分划索引的加权维纳多项式，且每个这样的终端都由齐次 q 重维纳积分实现。每个已表示的终端都有一个极小的可分分次支撑，其在分次正交变换下唯一。我们构造了一个正希尔伯特-斯坦提取，经过 J 步后，其残差高斯缺陷与界面缺陷为 O(J⁻¹)，而斯坦分解误差为 O(J⁻¹/²)，剩余的活性与协方差比较无有界能量速率。在齐次混沌中，格拉姆约化反馈收敛到可分解与本原福克投影，后者是典范独立高斯因子。所有边际四阶累积量的消失等价于可分解投影的消失，从而得到带有特征函数界的向量四阶矩定理。对于有限混合次数，按权重的三角过程消除了幽灵阻碍，恢复了所有单腿收缩检测到的极大高斯因子。
英文摘要
We study weak limits of multiple Wiener integrals together with the limits of their kernel contractions. After passage to a subsequence, bounded kernel sequences admit a Gaussian Hermite representation whose coordinates carry weights determined by the original chaos orders. In the limits of kernel contractions, only pairings of whole Gaussian coordinates of equal weight contribute. We construct kernel projections with explicit bounds for Gaussian approximation and asymptotic independence. For homogeneous inputs, spectral extraction converges to the decomposable and primitive projections in the limiting kernel spaces. Conversely, prescribed weighted Hermite expansions are realized by smooth compactly supported kernels away from the diagonals, with exact inner products for finite expansions and asymptotically prescribed contractions. We then classify fixed-chaos limits with an absolute exponential moment. For a standard Gaussian vector $X$ and a symmetric matrix $\mathsf A$, a quadratic target $X^{\mathsf T}\mathsf A X-Tr\mathsf A+b^{\mathsf T}X$ is such a limit if and only if $\mathsf A b=0$. We also give a finite cumulant criterion and a joint version. Finally, we identify limits under stationary Ornstein--Uhlenbeck noise: a coordinate of weight $s$ has relaxation rate $s$. Two fourth-chaos examples have identical one-time laws and covariance functions but different asymptotic variances for time averages of their squares. For kernels with nonnegative coordinates, contractions of the weight-one component along simple regular graphs are nonnegative; this excludes an explicit cubic representation that is attained by signed kernels.
