Weak limits of multiple Wiener integrals: representation, extraction, and realization

Abstract

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.