Periodicity is one of the most important characteristics of time series, and tests for periodicity go back to the very origins of the field. We consider the two situations where the potential period of a functional time series (FTS) is known and where it is unknown. For both problems we develop fully functional tests and work out the asymptotic distributions. When the period is known we allow for dependent noise and show that our test statistic is equivalent to the functional ANOVA statistic. The limiting distribution has an interesting form and can be written as a sum of independent hypoexponential variables whose parameters are eigenvalues of the spectral density operator of the FTS.
When the period is unknown our test statistic is based on the maximal norm of the functional periodogram over fundamental frequencies. The limiting distribution of this object is rather delicate: it requires a central limit theorem for vectors of functional data, where the number of components increases proportional to the sample size.
The talk is based on joint work with Piotr Kokoszka (Colorado State University) and Gilles Nisol (ULB) and Clément Cerovecki (ULB).