Abstract: We propose a new measure for stationarity in functional time series that is based on an explicit representation of the L?-distance between the spectral density operator of a nonstationary process and its best (L?-)approximation by a spectral density operator corresponding to a stationary process. This distance can be estimated by the sum of the Hilbert–Schmidt inner products of the periodogram operators (evaluated at different frequencies). Furthermore, the asymptotic normality of an appropriately standardized version of the estimator can be established for the corresponding estimator under the null and alternative hypotheses. As a result, we obtain a simple asymptotic frequency-domain level α-test (using the quantiles of the normal distribution) to test for the hypothesis of stationarity of a functional time series. We also briefly discuss other applications, such as asymptotic confidence intervals for the measure of stationarity, or the construction of tests for "relevant deviations from stationarity". We demonstrate in a small simulation study that the new method has very good finite-sample properties. Moreover, we apply our test to annual temperature curves.

Key words and phrases: Functional data, local stationarity, measuring stationarity, relevant hypotheses, spectral analysis, time series.