We apply the concept of distance covariance for testing independence of two long-range dependent time series. As test statistic we propose a linear combination of empirical distance cross-covariances. We derive the asymptotic distribution of the test statistic, and we show consistency against arbitrary alternatives. The asymptotic theory developed in this paper is based on a novel non-central limit theorem for stochastic processes with values in an $L^2$-Hilbert space. This limit theorem is of general theoretical interest which goes beyond the context of this article. Subject to the dependence in the data, the standardization and the limit distributions of the proposed test statistic vary. Since the limit distributions are unknown, we propose a subsampling procedure to determine the critical values for the proposed test, and we provide a proof for the validity of subsampling. In a simulation study, we investigate the finite-sample behavior of our test, and we compare its performance to tests based on the empirical cross-covariances. As an application of our results we analyze the cross-dependencies between mean monthly discharges of three rivers.

翻译：本文应用距离协方差概念检验两个长程依赖时间序列的独立性。作为检验统计量，我们提出了经验距离交叉协方差的线性组合。我们推导了该检验统计量的渐近分布，并证明了其对任意备择假设的一致性。本文发展的渐近理论基于一个新颖的$L^2$希尔伯特空间值随机过程的非中心极限定理。该极限定理具有超出本文范围的普适理论价值。根据数据的依赖结构，所提检验统计量的标准化及极限分布会发生变化。由于极限分布未知，我们提出一种子抽样程序来确定检验的临界值，并给出了子抽样有效性的证明。在模拟研究中，我们考察了检验的有限样本表现，并将其与基于经验交叉协方差的检验进行了性能对比。作为结果的应用，我们分析了三条河流月平均流量之间的交叉依赖关系。