This paper deals with analyzing structural breaks in the covariance operator of sequentially observed functional data. For this purpose, procedures are developed to segment an observed stretch of curves into periods for which second-order stationarity may be reasonably assumed. The proposed methods are based on measuring the fluctuations of sample eigenvalues, either individually or jointly, and traces of the sample covariance operator computed from segments of the data. To implement the tests, new limit results are introduced that deal with the large-sample behavior of vector-valued processes built from partial sample eigenvalue estimates. These results in turn enable the calibration of the tests to a prescribed asymptotic level. A simulation study and an application to Australian annual minimum temperature curves confirm that the proposed methods work well in finite samples. The application suggests that the variation in annual minimum temperature underwent a structural break in the 1950s, after which typical fluctuations from the generally increasing trendstarted to be significantly smaller.

中文翻译：

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协方差算子谱和迹的结构断裂分析

本文涉及分析顺序观察功能数据的协方差算子中的结构中断。为此，开发了程序将观察到的一段曲线分割成可以合理假设二阶平稳性的时期。所提出的方法基于单独或联合测量样本特征值的波动，以及从数据段计算的样本协方差算子的迹线。为了实施测试，引入了新的限制结果，以处理从部分样本特征值估计构建的向量值过程的大样本行为。这些结果反过来又使测试能够校准到规定的渐近水平。模拟研究和对澳大利亚年最低温度曲线的应用证实，所提出的方法在有限样本中运行良好。应用表明，年最低气温的变化在 1950 年代经历了结构性断裂，此后从普遍上升趋势的典型波动开始明显变小。