Breuer-Major Theorems for Hilbert Space-Valued Random Variables

TL;DR

This paper extends the Breuer-Major theorem to Hilbert space-valued Gaussian processes, establishing a CLT for transformed processes using Malliavin-Stein techniques, with applications in functional data analysis and neural operators.

Contribution

It introduces a Hilbert space-valued CLT for Gaussian processes with operator transformations, employing infinite-dimensional Malliavin-Stein methods, and provides new applications in neural operators.

Findings

Established a CLT for Hilbert space-valued Gaussian processes.

Provided quantitative and continuous-time versions of the CLT.

Applied results to neural operators and functional data analysis.

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.