  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. 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.
