Simultaneous Inference for Covariance and Precision Matrices of Long-Range Dependent Time Series

2026-04-17

AI Summary

This study addresses the challenge of inferring covariance and precision matrices from long-range dependent time series without imposing structural assumptions. The authors propose a finite-sample inference framework that integrates a Berry–Esseen-type Gaussian approximation, martingale and m-dependent approximations, and a novel triadic block construction, complemented by a block bootstrap procedure designed to preserve strong dependence structures. Theoretically, the work establishes non-asymptotic bounds on the infinity-norm estimation error of the covariance matrix, accommodating ultra-high-dimensional settings where the dimensionality grows sub-exponentially with the sample size. Notably, it extends valid statistical inference to the precision matrix for the first time under strong long-range dependence, ensuring reliable performance even in such complex dependency regimes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices.

Problem

• long-range dependence
• covariance matrix
• precision matrix
• simultaneous inference
• high-dimensional time series

Innovation

• long-range dependence
• Gaussian approximation
• triadic blocks
• block bootstrap
• ultra-high-dimensional