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.

翻译：受高维时间序列数据分析中统计推断问题的驱动，我们首先推导了高维相依随机向量之和在超矩形、简单凸集与稀疏凸集上高斯近似的非渐近误差界。我们研究了三种不同相依框架（α-混合、m-相依及物理相依度量）下时间依赖性对收敛到高斯随机向量速率的影响。特别地，我们在α-混合框架下建立了新的误差界，并在物理相依度量下推导出比现有结果更快的收敛速率。为将所得结果应用于实际统计推断问题，我们还基于长程协方差矩阵的核估计器推导了数据驱动的参数自助法程序。我们将统一的高斯与自助近似结果应用于时间序列数据的以下场景：基于ℓ²与ℓ∞型组合统计量的均值向量检验、变点检测，以及协方差矩阵与精度矩阵的置信域构建。