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Central limit theorems for high dimensional dependent data

Jinyuan Chang, Xiaohui Chen, Mingcong Wu

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 estimator for the long-run covariance matrices. We apply the unified Gaussian and bootstrap approximation results to test mean vectors with combined $\ell^2$ and $\ell^\infty$ type statistics, change point detection, and construction of confidence regions for covariance and precision matrices, all for time series data.

Central limit theorems for high dimensional dependent data

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 (-mixing, -dependent, and physical dependence measure). In particular, we establish new error bounds under the -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 estimator for the long-run covariance matrices. We apply the unified Gaussian and bootstrap approximation results to test mean vectors with combined and type statistics, change point detection, and construction of confidence regions for covariance and precision matrices, all for time series data.

Paper Structure

This paper contains 64 sections, 40 theorems, 360 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. High-dimensional central limit theorems
  3. High-dimensional CLT for hyper-rectangles
  4. $\alpha$-mixing sequence
  5. $m$-dependent sequence
  6. Sequence with physical dependence
  7. Comparison with existing result under physical dependence measure
  8. High-dimensional CLT for simple convex sets
  9. $\alpha$-mixing sequence
  10. $m$-dependent sequence
  11. Sequence with physical dependence
  12. High-dimensional CLT for sparsely convex sets
  13. $\alpha$-mixing sequence
  14. $m$-dependent sequence
  15. Sequence with physical dependence
  16. ...and 49 more sections

Key Result

Theorem \oldthetheorem

Assume $\{X_t\}$ is an $\alpha$-mixing sequence with $p\geqslant n^{\kappa}$ for some universal constant $\kappa>0$. Under Conditions as:tail--as:longrun, it holds that provided that $(\log p)^{3-\gamma_2}=o(n^{\gamma_2/3})$.

Theorems & Definitions (43)

  • Definition 1: $\alpha$-mixing coefficient
  • Theorem \oldthetheorem: Gaussian approximation for partial sums of the $\alpha$-mixing sequence
  • Theorem \oldthetheorem: Gaussian approximation for partial sums of a sequence under dependency graph
  • Corollary 1: Gaussian approximation for partial sums of an $m$-dependent sequence
  • Theorem \oldthetheorem: Gaussian approximation for maxima of partial sums of time series under functional dependence
  • Corollary 2: Overall rate of convergence under physical dependence
  • Proposition 1: Rate of convergence under physical dependence in ZhangWu_2017
  • Definition 2: Simple convex set
  • Theorem \oldthetheorem: Gaussian approximation for partial sums of the $\alpha$-mixing sequence for simple convex sets
  • Theorem \oldthetheorem: Gaussian approximation for partial sums of a sequence under dependency graph for simple convex sets
  • ...and 33 more