Breuer-Major Theorems for Hilbert Space-Valued Random Variables
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916251618508800 |
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| author | Düker, Marie-Christine Zoubouloglou, Pavlos |
| author_facet | Düker, Marie-Christine Zoubouloglou, Pavlos |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_11452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Breuer-Major Theorems for Hilbert Space-Valued Random Variables Düker, Marie-Christine Zoubouloglou, Pavlos Probability Statistics Theory 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. |
| title | Breuer-Major Theorems for Hilbert Space-Valued Random Variables |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2405.11452 |