Self-normalization has attracted considerable attention in the recent literature of time series analysis, but its scope of applicability has been limited to low-/fixed-dimensional parameters for low-dimensional time series. In this article, we propose a new formulation of self-normalization for inference about the mean of high-dimensional stationary processes. Our original test statistic is a U-statistic with a trimming parameter to remove the bias caused by weak dependence. Under the framework of nonlinear causal processes, we show the asymptotic normality of our U-statistic with the convergence rate dependent upon the order of the Frobenius norm of the long-run covariance matrix. The self-normalized test statistic is then constructed on the basis of recursive subsampled U-statistics and its limiting null distribution is shown to be a functional of time-changed Brownian motion, which differs from the pivotal limit used in the low-dimensional setting. An interesting phenomenon associated with self-normalization is that it works in the high-dimensional context even if the convergence rate of original test statistic is unknown. We also present applications to testing for bandedness of the covariance matrix and testing for white noise for high-dimensional stationary time series and compare the finite sample performance with existing methods in simulation studies. At the root of our theoretical arguments, we extend the martingale approximation to the high-dimensional setting, which could be of independent theoretical interest.

中文翻译：

---

通过自归一化对高维时间序列进行假设检验

自归一化在时间序列分析的最新文献中引起了相当大的关注，但是它的适用范围仅限于低维时间序列的低维/固定维参数。在本文中，我们提出了一种自归一化的新公式，用于推断高维平稳过程的均值。我们最初的检验统计量是带有修整参数的U统计量，用于消除因弱依赖性而引起的偏差。在非线性因果过程的框架下，我们显示了我们的U统计量的渐近正态性，其收敛速度取决于长期协方差矩阵的Frobenius范数的阶。然后，基于递归子采样的U统计量构造自归一化检验统计量，并显示其极限零分布是随时间变化的布朗运动的函数，该运动不同于低维设置中使用的枢轴极限。与自规范化相关的有趣现象是，即使原始测试统计量的收敛速度未知，它也可以在高维上下文中运行。我们还介绍了在高维平稳时间序列测试协方差矩阵的带度和白噪声测试中的应用，并将有限样本性能与模拟研究中的现有方法进行比较。在理论争论的基础上，我们将the近似法扩展到高维环境，这可能具有独立的理论意义。