Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density operators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying functional time series. The properties of the test are derived both under the null hypothesis of stationary functional time series and under the smooth alternative of locally stationary functional time series. The methodology is theoretically justified through asymptotic results. Evidence from simulation studies and an application to annual temperature curves suggests that the test works well in finite samples.

中文翻译：

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在频域中测试函数时间序列的平稳性

近来，人们对函数时间序列的兴趣激增，涵盖方法论和应用程序的论文以更快的速度发表。本文提出了一种新的基于频域方法的函数时间序列的平稳性检验，为该领域的研究做出了贡献。建议的测试统计基于通过跨所有傅立叶频率的谱密度算子的函数主成分分析的联合降维，明确允许与频率相关的截断水平以适应基础函数时间序列的动态。检验的性质是在平稳函数时间序列的原假设和局部平稳函数时间序列的平滑替代下得出的。该方法通过渐近结果在理论上是合理的。来自模拟研究和年温度曲线应用的证据表明，该测试在有限样本中运行良好。