A time-domain test for the assumption of second-order stationarity of a functional time series is proposed. The test is based on combining individual cumulative sum tests which are designed to be sensitive to changes in the mean, variance and autocovariance operators, respectively. The combination of their dependent p values relies on a joint-dependent block multiplier bootstrap of the individual test statistics. Conditions under which the proposed combined testing procedure is asymptotically valid under stationarity are provided. A procedure is proposed to automatically choose the block length parameter needed for the construction of the bootstrap. The finite-sample behavior of the proposed test is investigated in Monte Carlo experiments, and an illustration on a real data set is provided.

中文翻译：

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检测局部平稳函数时间序列中二阶平稳性的偏差

提出了对函数时间序列二阶平稳性假设的时域测试。该测试基于组合单独的累积总和测试，这些测试旨在分别对均值、方差和自协方差运算符的变化敏感。它们的相关 p 值的组合依赖于单个测试统计数据的联合相关块乘数引导程序。提供了建议的组合测试程序在平稳性下渐近有效的条件。提出了一个程序来自动选择构建引导程序所需的块长度参数。在蒙特卡罗实验中研究了所提出测试的有限样本行为，并提供了真实数据集的说明。