Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let \Xₖ\₊ ₙ be a stationary Gaussian process with values in a separable Hilbert space H₁, and let G: H₁ H₂ be an operator acting on Xₖ. Under suitable conditions on the operator G and the temporal and cross-sectional correlations of \Xₖ\₊ ₙ, we derive a central limit theorem (CLT) for the normalized partial sums of \GXₖ\₊ ₙ. To prove a CLT for the Hilbert space-valued process \GXₖ\₊ ₙ, we employ techniques from the recently developed infinite dimensional Malliavin-Stein framework. In addition, we provide quantitative and continuous time versions of the derived CLT. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present a novel limit theorem in the framework of neural operators as an application of our result.