Locally Stationary Functional Time Series

The literature on time series of functional data has focused on pro- cesses of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the func- tional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for funda- mental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are inves- tigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cramér representation for an impor- tant class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.