A statistical hypothesis test for long range dependence (LRD) in functional time series in manifolds has been formulated in Ruiz-Medina and Crujeiras (2025) in the spectral domain for fully observed functional data. The asymptotic Gaussian distribution of the proposed test statistics, based on the weighted periodogram operator, under the null hypothesis, and the consistency of the test have been derived. In this paper, we analyze the asymptotic properties of this spectral LRD testing procedure, when functional data are contaminated, and discretely observed through random uniform spatial sampling.

翻译：Ruiz-Medina与Crujeiras（2025）在谱域中针对完全观测的函数数据，提出了流形中函数时间序列长程依赖性（LRD）的统计假设检验。基于加权周期图算子的检验统计量在原假设下的渐近高斯分布及检验的一致性已得到推导。本文分析了当函数数据受到污染且通过随机均匀空间采样进行离散观测时，该谱域LRD检验方法的渐近性质。