February 2024 Central limit theorems for high dimensional dependent data
Jinyuan Chang, Xiaohui Chen, Mingcong Wu
Author Affiliations +
Bernoulli 30(1): 712-742 (February 2024). DOI: 10.3150/23-BEJ1614

Abstract

Motivated by statistical inference problems in high-dimensional time series data analysis, we first derive non-asymptotic error bounds for Gaussian approximations of sums of high-dimensional dependent random vectors on hyper-rectangles, simple convex sets and sparsely convex sets. We investigate the quantitative effect of temporal dependence on the rates of convergence to a Gaussian random vector over three different dependency frameworks (α-mixing, m-dependent, and physical dependence measure). In particular, we establish new error bounds under the α-mixing framework and derive faster rate over existing results under the physical dependence measure. To implement the proposed results in practical statistical inference problems, we also derive a data-driven parametric bootstrap procedure based on a kernel-type estimator for the long-run covariance matrices. The unified Gaussian and parametric bootstrap approximation results can be used to test mean vectors with combined 2 and type statistics, do change point detection, and construct confidence regions for covariance and precision matrices, all for time series data.

Funding Statement

Chang and Wu were supported in part by the National Natural Science Foundation of China (grant nos. 71991472, 72125008 and 11871401). Chang was also supported by the Center of Statistical Research at Southwestern University of Finance and Economics. Chen was supported by in part by the National Science Foundation (grant no. 1752614).

Acknowledgments

The authors would like to thank the editor, the associate editor and two reviewers for their constructive suggestions which led to the improvements of the paper.

Citation

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Jinyuan Chang. Xiaohui Chen. Mingcong Wu. "Central limit theorems for high dimensional dependent data." Bernoulli 30 (1) 712 - 742, February 2024. https://doi.org/10.3150/23-BEJ1614

Information

Received: 1 June 2022; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665595
zbMATH: 07788901
Digital Object Identifier: 10.3150/23-BEJ1614

Keywords: central limit theorem , dependent data , Gaussian approximation , high-dimensional statistical inference , Parametric bootstrap

Vol.30 • No. 1 • February 2024
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