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.

翻译：设$\{X_k\}_{k \in \mathbb{Z}}$为取值于可分Hilbert空间$\mathcal{H}_1$的平稳高斯过程，$G:\mathcal{H}_1 \to \mathcal{H}_2$为作用于$X_k$的算子。在算子$G$以及$\{X_k\}_{k \in \mathbb{Z}}$的时间与截面相关性满足适当条件的假设下，我们推导了$\{G[X_k]\}_{k \in \mathbb{Z}}$标准化部分和序列的中心极限定理。为证明该Hilbert空间值过程$\{G[X_k]\}_{k \in \mathbb{Z}}$的CLT，我们采用了近期发展的无穷维Malliavin-Stein框架技术。此外，我们给出了所导出CLT的定量版本与连续时间版本。通过一系列实例，我们恢复并强化了函数型数据分析中若干重要统计量的极限定理，并以神经算子框架中的新型极限定理作为我们结果的应用展示。