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Breuer-Major Theorems for Hilbert Space-Valued Random Variables
Published 19 May 2024 in math.PR, math.ST, and stat.TH | (2405.11452v1)
Abstract: Let ${X_k}{k \in \mathbb{Z}}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal{H}_1$, and let $G:\mathcal{H}_1 \to \mathcal{H}_2$ be an operator acting on $X_k$. Under suitable conditions on the operator $G$ and the temporal and cross-sectional correlations of ${X_k}{k \in \mathbb{Z}}$, we derive a central limit theorem (CLT) for the normalized partial sums of ${G[X_k]}{k \in \mathbb{Z}}$. To prove a CLT for the Hilbert space-valued process ${G[X_k]}{k \in \mathbb{Z}}$, 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.
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