Central limit theorems for high dimensional dependent data

Understand

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 ($\alpha$-mixing, $m$-dependent, and physical dependence measure).
• In particular, we establish new error bounds under the $\alpha$-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.