High-dimensional Gaussian Graphical Model Testing for Long-Memory Time Series

2026-09-24

AI Summary

This study addresses the theoretical gaps and technical bottlenecks in conditional independence testing for long-memory, high-dimensional time series. It proposes a data-adaptive statistic to evaluate conditional independence within graph structures, establishing Berry-Esseen-type Gaussian approximation bounds applicable to both short- and long-memory processes. Furthermore, a consistent correction procedure is developed to accommodate ultrahigh-dimensional settings. By integrating the block bootstrap with finite-sample inference techniques, the proposed framework achieves fully data-adaptive testing. This work fills a critical void in the literature by offering a method that simultaneously ensures asymptotic consistency and high statistical power. Its practical utility is effectively validated through functional magnetic resonance imaging (fMRI) brain functional connectivity analysis, demonstrating robust performance in real-world neuroimaging applications.

Abstract

Many real-world high-dimensional time series exhibit long-memory, but Gaussian graphical model testing in this regime remains understudied. We develop a direct, data-adaptive test statistic for assessing conditional independence in the graph structure of stationary Gaussian time series. We establish a finite-sample, Berry--Esseen type Gaussian approximation bound for the statistic, which applies to both short-memory and long-memory time series. The testing procedure is fully data-adaptive using block bootstrap method, on which we provide a finite-sample validity result including in the ultra-high-dimensional scenario, and can be extended to comparing graphical structures in two-sample tests. We also develop a consistency-empowered correction to the statistic and show that such tests attain asymptotic consistency in both size and power. Our proposed method is applied to a real-world fMRI data to understand functional connectivities within brain in different periods.

Problem

• High-dimensional
• Gaussian Graphical Model
• Long-memory time series
• Conditional independence testing

Innovation

• High-dimensional Gaussian Graphical Model
• Long-memory Time Series
• Data-adaptive Testing
• Block Bootstrap
• Berry-Esseen Bound