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 finitesample behavior of the proposed test is investigated in Monte Carlo experiments and an illustration on a real data set is provided.

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Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly non-stationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be non-relevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model Xi = µ(i/n) + εi with possibly non-stationary errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function µ from a functional g(µ) (such as the value µ(0) or the integral 1 0 µ(t)dt) is smaller than a given threshold on a given interval [x0, x1] ⊆ [0, 1]. A test for this type of hypotheses is developed using an appropriate estimator, say d∞,n, for the maximum deviation d∞ = sup t∈[x 0 ,x 1 ] |µ(t) − g(µ)|. We derive the limiting distribution of an appropriately standardized version of d∞,n, where the standardization depends on the Lebesgue measure of the set of extremal points of the function µ(•) − g(µ). A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example.

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