  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 ($\alpha$-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-type estimator for the long-run
covariance matrices.
