Temporally correlated error process is commonly encountered in practice and poses significant challenges in high-dimensional statistical analysis. This paper conducts low-dimensional inference for high-dimensional linear models with stationary errors. We adopt the framework of functional dependence measure for adequate accommodation of the error correlation. A new desparsifying Lasso based testing procedure is developed by incorporating a banded estimator of the error autocovariance matrix. Asymptotic normality of the proposed estimator is established by demonstrating the consistency of the banded autocovariance matrix estimator. The result indicates how the range of p is substantially narrower if the moment condition of error weakens or the dependence becomes stronger. We further develop a data-driven choice of the banding parameter. The simulation studies illustrate the satisfactory finite-sample performance of our proposed procedure, and a real data example is also presented for illustration.

在实践中通常会遇到与时间相关的错误过程，这在高维统计分析中提出了重大挑战。本文针对具有固定误差的高维线性模型进行低维推断。我们采用功能依赖度量的框架，以适当地容纳误差相关性。通过合并误差自协方差矩阵的带状估计器，开发了一种新的基于Lasso的测试程序。通过证明带状自协方差矩阵估计器的一致性来建立所提出估计器的渐近正态性。结果表明p的范围如果错误时刻的条件变弱或相关性变强，则α会大大变窄。我们进一步开发了带区参数的数据驱动选择。仿真研究说明了我们提出的程序的令人满意的有限样本性能，并且还提供了一个实际的数据示例来进行说明。