This manuscript studies the problem of independence testing between two high-dimensional time series without assuming weak stationarity, that is, allowing their autocovariances to vary over time. To this end, we propose a bimodal weighted-average test statistic that removes the bias induced by temporal dependence under the null hypothesis, thereby avoiding the need to whiten the time series prior to hypothesis testing -- a procedure that is challenging in high-dimensional and nonstationary settings. To facilitate statistical inference, we develop a dependent wild bootstrap procedure. On the theoretical side, we derive a concentration inequality for quadratic forms of time series data stemming from a class of high-dimensional, nonlinear, and nonstationary processes. This result enables us to derive the asymptotic null distribution of the proposed test statistic and to establish the validity of the bootstrap algorithm. Numerical results show that the proposed test attains desired size and good power performance even when the dimension exceeds the sample size or when the data-generating process exhibits time-varying autocovariances. In contrast, tests based on whitening time series fail to maintain correct size in the presence of unstable autocovariance structures. Since nonstationary autocovariances commonly arise in real-life time series data, our work offers a robust procedure for independence testing.

翻译：本文研究两个高维时间序列之间的独立性检验问题，不假设弱平稳性，即允许其自协方差随时间变化。为此，我们提出一种双模态加权平均检验统计量，该统计量在原假设下消除了时间依赖性带来的偏差，从而避免在假设检验前对时间序列进行白化处理——这一过程在高维和非平稳场景中具有挑战性。为促进统计推断，我们开发了一种依赖型野刀自举（bootstrap）程序。在理论方面，我们针对来自一类高维、非线性及非平稳过程的时间序列数据的二次型，推导出一个浓度不等式。该结果使我们能够推导出所提检验统计量的渐近零分布，并证明自举算法的有效性。数值结果表明，即使当维数超过样本量或数据生成过程呈现时变自协方差时，所提检验仍能达到理想的检验水平并具有良好的功效性能。相比之下，基于白化时间序列的检验在存在不稳定自协方差结构时无法维持正确的检验水平。由于非平稳自协方差在现实时间序列数据中普遍存在，我们的工作为独立性检验提供了一种稳健的程序。