A statistical hypothesis test for long range dependence (LRD) in manifold-supported functional time series is formulated in the spectral domain. The proposed test statistic is based on the weighted periodogram operator, assuming that the elements of the spectral density operator family are invariant with respect to the group of isometries of the manifold. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistics operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is established under the alternative hypothesis. The consistency of the test is then derived, from the obtained consistency of the integrated weighted periodogram operator under LRD. Practical implementation of our testing approach is based on the random projection methodology. The frequency-varying Karhunen-Lo\'eve expansion of invariant Gaussian random spectral Hilbert-Schmidt kernels on manifolds is considered for generation of random directions in the implementation of this methodology. A simulation study illustrates the main results regarding asymptotic normality and consistency, and the empirical size and power properties of the proposed testing approach.

翻译：本文在谱域中构建了一种针对流形支撑函数时间序列长程依赖性(LRD)的统计假设检验方法。所提出的检验统计量基于加权周期图算子，并假设谱密度算子族元素关于流形等距群保持不变。通过推导中心极限定理，获得了零假设下该检验统计算子的渐近高斯分布。在备择假设下，建立了积分经验二阶与四阶累积量谱密度算子偏差在希尔伯特-施密特算子范数意义下趋于零的收敛速率。基于所得LRD条件下积分加权周期图算子的一致性，进而推导出该检验的一致性。本检验方法的实际实施基于随机投影方法，其中通过考虑流形上不变高斯随机谱希尔伯特-施密特核的频率变分Karhunen-Loéve展开来生成该方法实施过程中的随机方向。模拟研究展示了关于渐近正态性与一致性的主要结果，以及所提出检验方法的经验尺度与功效特性。