Central limit theorems for high dimensional dependent data

Jinyuan Chang, Xiaohui Chen, Mingcong Wu

Research output: Contribution to journalArticlepeer-review


Motivated by statistical inference problems in high-dimensional time series data analysis, we first derive nonasymptotic 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 ℓ2 and ℓ type statistics, do change point detection, and construct confidence regions for covariance and precision matrices, all for time series data.

Original languageEnglish (US)
Pages (from-to)712-742
Number of pages31
Issue number1
StatePublished - Feb 2024


  • Central limit theorem
  • Gaussian approximation
  • dependent data
  • high-dimensional statistical inference
  • parametric bootstrap

ASJC Scopus subject areas

  • Statistics and Probability


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