Towards Frequency Domain Analysis of Stationary Functional Data

We consider the problem of drawing statistical inferences on the second-order structure of weakly dependent functional time series. When functional data are independent, the entire second order structure is captured by the covariance operator. For dependent functional data, one needs to consider covariance operators relating different lags of the series, as is the case in multivariate time series. In the functional case, most work has focused on inference for stationary time series that are linear - a case that is now well understood. More recent work has focused on the estimation of the mean, long-run covariance operator and principal components on the basis of moment type mixing conditions for time series that are not necessarily linear. In this talk we consider the problem of inferring the complete second-order structure of stationary functional time series without a priori structural modeling assumptions. Our approach is to formulate a frequency domain framework for weakly dependent functional data, employing suitable generalisations of finite-dimensional notions. We introduce the basic ingredients of such a framework, propose estimators, and study their asymptotics under functional cumulant-type mixing conditions.
(Based on joint work with Shahin Tavakoli, EPFL).