Central Limit Theorems for High Dimensional Dependent Data
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Motivated by statistical inference problems in high-dimensional time series analysis, we 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 normality 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.
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