This article studies bootstrap inference for high dimensional weakly dependent time series in a general framework of approximately linear statistics. The following high dimensional applications are covered: (1) uniform confidence band for mean vector; (2) specification testing on the second order property of time series such as white noise testing and bandedness testing of covariance matrix; (3) specification testing on the spectral property of time series. In theory, we first derive a Gaussian approximation result for the maximum of a sum of weakly dependent vectors, where the dimension of the vectors is allowed to be exponentially larger than the sample size. In particular, we illustrate an interesting interplay between dependence and dimensionality, and also discuss one type of "dimension free" dependence structure. We further propose a blockwise multiplier (wild) bootstrap that works for time series with unknown autocovariance structure. These distributional approximation errors, which are finite sample valid, decrease polynomially in sample size. A non-overlapping block bootstrap is also studied as a more flexible alternative. The above results are established under the general physical/functional dependence framework proposed in Wu (2005). Our work can be viewed as a substantive extension of Chernozhukov et al. (2013) to time series based on a variant of Stein's method developed therein.
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