A Moment-Based Notion of Time Dependence for Functional Time Series

Abstract

This paper addresses the fundamental topic of time dependence for time series when data points are given as functions. We construct a notion of time dependence through the scores of the principal components, which allows us to adapt various scalar time series techniques to the functional data context. In particular, we define dependence based on the autocovariances and cumulants of the scores, covering short and long memory scenarios. This notion naturally applies to linear processes. To justify this moment based approach we investigate the asymptotic properties of the estimator of the functional principal components and show its consistency under short and long memory. Finally, the applicability of our notion is illustrated through several statistical problems in functional time series: estimation of the functional autoregressive model, estimation of the long-run covariance function and testing for short memory against the long memory alternative.