The development of high-dimensional white noise test is important in both statistical theories and applications, where the dimension of the time series can be comparable to or exceed the length of the time series. This paper proposes several distribution-free tests using the rank based statistics for testing the high-dimensional white noise, which are robust to the heavy tails and do not quire the finite-order moment assumptions for the sample distributions. Three families of rank based tests are analyzed in this paper, including the simple linear rank statistics, non-degenerate U-statistics and degenerate U-statistics. The asymptotic null distributions and rate optimality are established for each family of these tests. Among these tests, the test based on degenerate U-statistics can also detect the non-linear and non-monotone relationships in the autocorrelations. Moreover, this is the first result on the asymptotic distributions of rank correlation statistics which allowing for the cross-sectional dependence in high dimensional data.

翻译：高维白噪音测试的发展在统计理论和应用中都很重要,时间序列的维度可以与时间序列的长度可比或超过时间序列的长度。本文件建议使用基于等级的统计进行若干次无分布式测试,以测试高维的白色噪音,这些噪音对重尾是坚固的,对样本分布的定序时假设并不重要。本文分析了基于等级的三组测试,包括简单的线性级统计、非变性U-统计学和退化的U-统计学。为这些测试的每个家庭确定了无症状分布和速率的最佳性。在这些测试中,基于堕落的U-统计学的测试还可以探测到在自动关系中的非线性和非分子关系。此外,这是基于等级相关性统计的无症状分布的第一个结果,允许在高维数据中出现交叉依赖性。