Weak Limits of Wiener Chaos: Primitive-Fock Classification and Hilbert-Stein Extraction

Abstract

We characterize the weak closure of uniformly $L^2$-bounded vectors in a fixed Wiener chaos when the underlying Gaussian Hilbert spaces may vary. On a represented subsequence, each physical-weight tensor space splits into a decomposable closed span and its primitive orthogonal complement. Primitive-block evaluation extends to a unitary weighted Fock representation. Hence the weak limits of a $q$th chaos are exactly the weighted Wiener polynomials indexed by partitions of $q$, and every such terminal is realized by homogeneous $q$-fold Wiener integrals. Each represented terminal has a minimal separable graded support, unique up to graded orthogonal transformations. We construct a positive Hilbert-Stein extraction. After $J$ steps, its residual Gaussian and interface defects are $O(J^{-1})$, while the Stein factorization error is $O(J^{-1/2})$. The remaining active and covariance comparisons have no bounded-energy rate. In homogeneous chaos, Gram-reduced feedback converges to the decomposable and primitive Fock projections; the latter is the canonical independent Gaussian factor. The vanishing of all marginal fourth cumulants is equivalent to disappearance of the decomposable projection, recovering the vector fourth-moment theorem with a characteristic-function bound. For finite mixed degrees, a weightwise triangular procedure removes the ghost obstruction and recovers the maximal Gaussian factor detected by all one-leg contractions.